Linear Equation
•Download as PPT, PDF•
2 likes•1,593 views
This document discusses the relationship between linear algebraic expressions and vectors. A linear algebraic expression consists of two or more linear algebraic terms combined using addition, subtraction, or both. Vectors can be represented using linear algebraic expressions, as demonstrated by an example involving a triangle where different vectors are written in terms of other vectors a and b. The document notes that algebraic expressions are commonly used in vectors, and this example is just one of several connections between linear equations and vectors.
Report
Share
Report
Share
![[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]](https://image.slidesharecdn.com/tmp20233-100731063850-phpapp02/85/Linear-Equation-1-320.jpg)
![[object Object],[object Object],[object Object]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Recommended
Linear Equations
This document discusses the relationship between linear algebraic expressions and vectors. A linear algebraic expression consists of two or more linear algebraic terms combined using addition, subtraction, or both. An example problem involves a triangle PQR with midpoint S, where PS = 4a and PR = 3b. Students are asked to write vectors in terms of a and b, showing how linear algebraic expressions are used to represent vectors. The document notes that many vector problems involve algebraic expressions, demonstrating the connection between linear equations and vectors.
Topic 3
This document discusses the relationship between algebraic expressions and vectors. Algebraic expressions are used to solve problems involving addition and subtraction of vectors, where vectors are expressed in terms of other vectors. An example problem involves a triangle PQR where points T and S are given on sides RQ and PQ respectively using ratios relating distances. The vectors P, S, Q, T, and R must be determined in terms of other vectors using algebraic expressions.
Pythagorean theorem slide
The document explains the Pythagorean theorem and distance formula. The Pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides. It provides an example of 3^2 + (-4)^2 = c^2. The distance formula gives the distance between pairs of points based on their coordinates, and is based on the Pythagorean theorem. It provides an example of calculating the distance between points (2, -4) and (2, 10).
1.2 Ruler Postulates
The document discusses three geometric postulates:
1) The ruler postulate establishes a one-to-one correspondence between points on a line and real numbers on the number line, where the distance between points equals the absolute value of the difference of their corresponding numbers.
2) The ruler placement postulate allows choosing a number line such that two given points correspond to 0 and a positive number.
3) The segment addition postulate states that if one point is between two others, the sum of the distances to the end points equals the distance between the outer points.
Proving lines parallel
This document discusses proving that two lines are parallel using angles formed by a transversal. It provides examples of using the converse of corresponding angles, alternate interior angles, same-side interior angles, and vertical angles postulates and theorems to show parallel lines. The document also includes warm up questions about converses, examples proving lines parallel, and a quiz to assess understanding.
1 5 measuring segments
This document provides instruction on measuring segment lengths using rulers and postulates. It explains the ruler postulate, which relates the distance between points on a line to real numbers, and the segment addition postulate, which states that the sum of two segments with a shared endpoint is equal to the third segment between the outer points. Examples are provided to demonstrate using these postulates to find unknown segment lengths. The midpoint postulate is also introduced, relating a segment to the two segments formed by its midpoint.
Cal 3
This lecture discusses distance, midpoint, slope, lines, symmetries of graphs, equations of circles, and quadratic equations. It defines distance as the square root of the sum of the squared differences of x- and y-coordinates between two points. The midpoint formula finds the midpoint of a line segment between two points. Slope is defined as the rise over the run between two points on a line. Lines can be written in point-slope form, slope-intercept form, and intercept forms. Parallel and perpendicular lines are identified based on equal or negative reciprocal slopes. Symmetries of graphs include reflections across the x-axis, y-axis, or origin. The equation of a circle is given by (x-h)2
COORDINATE GEOMETRY
Coordinate geometry is the study of representing geometric figures on two- or three-dimensional planes. Some key concepts in coordinate geometry include:
- Finding the distance between two points using the Pythagorean theorem.
- Finding the midpoint of a line segment by taking the average of the x- and y-coordinates of the two endpoints.
- Dividing a line segment into parts externally or internally based on a given ratio of distances.
- Calculating the gradient or slope of a line as the ratio of vertical to horizontal change between two points on the line.
- Determining whether three points are collinear by checking if they have the same slope.
Recommended
Linear Equations
This document discusses the relationship between linear algebraic expressions and vectors. A linear algebraic expression consists of two or more linear algebraic terms combined using addition, subtraction, or both. An example problem involves a triangle PQR with midpoint S, where PS = 4a and PR = 3b. Students are asked to write vectors in terms of a and b, showing how linear algebraic expressions are used to represent vectors. The document notes that many vector problems involve algebraic expressions, demonstrating the connection between linear equations and vectors.
Topic 3
This document discusses the relationship between algebraic expressions and vectors. Algebraic expressions are used to solve problems involving addition and subtraction of vectors, where vectors are expressed in terms of other vectors. An example problem involves a triangle PQR where points T and S are given on sides RQ and PQ respectively using ratios relating distances. The vectors P, S, Q, T, and R must be determined in terms of other vectors using algebraic expressions.
Pythagorean theorem slide
The document explains the Pythagorean theorem and distance formula. The Pythagorean theorem states that the square of the hypotenuse of a right triangle equals the sum of the squares of the other two sides. It provides an example of 3^2 + (-4)^2 = c^2. The distance formula gives the distance between pairs of points based on their coordinates, and is based on the Pythagorean theorem. It provides an example of calculating the distance between points (2, -4) and (2, 10).
1.2 Ruler Postulates
The document discusses three geometric postulates:
1) The ruler postulate establishes a one-to-one correspondence between points on a line and real numbers on the number line, where the distance between points equals the absolute value of the difference of their corresponding numbers.
2) The ruler placement postulate allows choosing a number line such that two given points correspond to 0 and a positive number.
3) The segment addition postulate states that if one point is between two others, the sum of the distances to the end points equals the distance between the outer points.
Proving lines parallel
This document discusses proving that two lines are parallel using angles formed by a transversal. It provides examples of using the converse of corresponding angles, alternate interior angles, same-side interior angles, and vertical angles postulates and theorems to show parallel lines. The document also includes warm up questions about converses, examples proving lines parallel, and a quiz to assess understanding.
1 5 measuring segments
This document provides instruction on measuring segment lengths using rulers and postulates. It explains the ruler postulate, which relates the distance between points on a line to real numbers, and the segment addition postulate, which states that the sum of two segments with a shared endpoint is equal to the third segment between the outer points. Examples are provided to demonstrate using these postulates to find unknown segment lengths. The midpoint postulate is also introduced, relating a segment to the two segments formed by its midpoint.
Cal 3
This lecture discusses distance, midpoint, slope, lines, symmetries of graphs, equations of circles, and quadratic equations. It defines distance as the square root of the sum of the squared differences of x- and y-coordinates between two points. The midpoint formula finds the midpoint of a line segment between two points. Slope is defined as the rise over the run between two points on a line. Lines can be written in point-slope form, slope-intercept form, and intercept forms. Parallel and perpendicular lines are identified based on equal or negative reciprocal slopes. Symmetries of graphs include reflections across the x-axis, y-axis, or origin. The equation of a circle is given by (x-h)2
COORDINATE GEOMETRY
Coordinate geometry is the study of representing geometric figures on two- or three-dimensional planes. Some key concepts in coordinate geometry include:
- Finding the distance between two points using the Pythagorean theorem.
- Finding the midpoint of a line segment by taking the average of the x- and y-coordinates of the two endpoints.
- Dividing a line segment into parts externally or internally based on a given ratio of distances.
- Calculating the gradient or slope of a line as the ratio of vertical to horizontal change between two points on the line.
- Determining whether three points are collinear by checking if they have the same slope.
Calculus
This lecture discusses inequalities, absolute values, and graphs of solutions to inequalities on a coordinate plane. Key points covered include:
- Inequalities relate values that are different using symbols like < and >. Operations on inequalities follow rules like adding/subtracting a positive number or multiplying/dividing by a negative number requires changing the inequality symbol.
- Absolute value expressions use | | to represent the distance from zero. Operations inside | | follow different rules than normal order of operations.
- Graphing solutions to inequalities involves open/closed circles and drawing lines left/right to represent <, >, ≤, or ≥. Examples are worked through to demonstrate graphing linear and quadratic inequalities.
11.4 power point
The document discusses formulas for finding distance and midpoints between points on a coordinate plane. It provides examples of using the distance formula to calculate distances between points and the Pythagorean theorem to find the length of a hypotenuse. It also demonstrates using the midpoint formula, which takes the average of the x-values and y-values of two points to find the midpoint between them. This is then used along with distance calculations to solve problems involving finding distances or midpoints on a map.
GH 2-6
This document outlines the structure of a mathematical proof and provides an example proof of Theorem 2-7.
The structure of a proof consists of 5 parts: 1) stating the theorem, 2) providing a diagram of given information, 3) listing what is given, 4) listing what is to be proved, 5) providing statements and reasons that lead from the given to the conclusion.
The example proof of Theorem 2-7 states that if two angles are supplements of congruent angles, then the two angles are congruent. The proof uses 5 statements and reasons: 1) stating what is given, 2) defining supplementary angles, 3) substituting equal expressions, 4) using one given fact, 5) applying
Planning a proof
The document outlines the typical structure for proving a theorem, which consists of 5 parts: 1) stating the theorem, 2) illustrating it with a diagram, 3) listing what is given, 4) listing what is to be proved, and 5) a series of statements and reasons leading from the givens to the conclusion. It then provides an example proof of the theorem: "If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent." The proof uses the standard structure and logical reasoning to show that if two pairs of angles are supplementary, then the corresponding angles are congruent.
Mathematical skills II - Rearranging Equations
The document discusses rearranging formulas by making a different variable the subject of the formula. It explains that the process is similar to solving equations, where the same quantity is added or subtracted to both sides, or both sides are multiplied or divided by the same quantity. Examples are provided of rearranging formulas by changing the operations to make a specified variable the subject. The practice problems ask the reader to rearrange several formulas by making the variable listed at the end the subject of each formula.
Reflection of a point about a line & a plane in 2-D & 3-D co-ordinate systems...
Reflection of a point about a line & a plane in 2-D & 3-D co-ordinate systems...Harish Chandra Rajpoot
The document discusses methods for calculating the reflection of points about lines and planes in 2D and 3D coordinate systems. It provides general expressions to find the coordinates of a reflected point given a point and line/plane of reflection. For a point reflected about a line or plane, the midpoint between the original and reflected point must lie on the line/plane and the line connecting the points must be perpendicular/parallel to the line/plane. Using geometry relationships, the author derives formulas to calculate the coordinates of the reflected point and foot of the perpendicular from the original point to the line/plane.Best guess list for cal paper foundation
This document provides a summary of topics that may appear on the Edexcel Foundation Paper 2 exam, including:
1) Finding the lowest common multiple and highest common factor of two numbers.
2) Expressing numbers as products of primes, ordering fractions, decimals and percentages, and converting between them.
3) Solving ratio problems in context and expressing numbers as percentages of other numbers.
Right triangle similarity
This document provides instruction on similarity in right triangles. It begins with examples of identifying similar right triangles and finding geometric means. Geometric mean is defined as the positive square root of the product of two numbers. The document then shows how geometric means can be used to find unknown side lengths in right triangles using proportions. Examples demonstrate applications to measurement problems. The lesson concludes with a quiz reviewing key concepts like writing similarity statements and using proportions with geometric means to solve for missing side lengths in right triangles.
Similarity on right triangle
Similarity on a Right Triangle
Right Triangle with altitude
Right Triangle Theorem
30-60-90 Triangle Theorem
Isosceles Right Triangle Theorem
2.6 prove statements about segments and angles
This document contains examples and explanations of geometric proofs involving segments, angles, and properties of equality and congruence. It includes examples proving statements about segments and angles being equal based on given information. One example proves two statements: (1) that the distance between the food court and music store is the same as the distance between the shoe store and bookstore, and (2) that this is true because the music store is halfway between the food court and shoe store. The document provides guidance on writing two-column geometric proofs and practicing proofs involving segment and angle properties.
2.7 prove angle pair relationships
This document contains examples and explanations of angle pair relationships and angle proofs. It begins with definitions of equal angles if they are both right angles, angles formed by perpendicular lines, and congruent segment angles. Several examples of angle proofs are provided using statements and reasons. Guided practice problems ask to find missing angle measures using angle relationships. The document concludes with an exit slip and homework assignment.
Unit 1. day 17 studyguide1
This document contains various sections on geometry topics including working warm-up problems about writing equations involving angle pairs formed by parallel lines cut by a transversal. It also lists common core standards about transformations of geometric figures and using properties of parallel lines cut by transversals to establish facts about angles. Finally, it reminds students that their exam is on Thursday and includes a study guide they should complete without talking.
Geom9point7 97
This document discusses vectors and their components, magnitude, direction, and addition. It provides examples of calculating the magnitude and component form of vectors. It also explains how to add vectors using the parallelogram method by drawing a parallelogram and finding the diagonal vector sum, or the component method by finding the horizontal and vertical components and adding them. Vector addition is important in physics for the law of conservation of momentum.
Roots of polynomials
The Bairstow method and Muller method are techniques for calculating the roots of polynomials.
The Bairstow method uses synthetic division to iteratively calculate the roots of a polynomial by dividing it by quadratic factors (x^2 - rx - s) until the remainder is zero. It obtains better approximations of r and s at each iteration to isolate the roots.
The Muller method fits a parabola through three initial guesses to obtain coefficients a, b, c. It then uses the quadratic formula on the parabola to find the root, and iterates with new guesses until converging on a solution.
Both methods use iterative techniques to refine initial guesses and isolate the roots of polynomials without using
Factoring
This document provides an overview of different factoring techniques for quadratics, including:
1) Looking for patterns in the terms like perfect squares or a variable being cubed to use shortcuts.
2) Using the greatest common factor when there are only two terms or the coefficient of x is greater than 1.
3) Factoring the difference of squares or perfect square trinomials using their respective shortcuts.
4) Completely factoring polynomials using the generic rectangle/diamond method or variations for more complex expressions.
Math unit38 vectors
This document contains 6 presentations on vectors:
1) Identifying equivalent and opposite vectors in a diagram
2) Calculating components of vectors
3) Writing vector expressions in terms of other vectors
4) Illustrating vector addition and subtraction on a grid
5) Expressing vectors in terms of other vectors using properties of midpoints
6) Expressing vectors in terms of other vectors and proving collinearity using properties of parallelograms
Closest point on line segment to point
This document discusses two methods for finding closest points between line segments and points or between two lines:
1) To find the closest point on a line segment to a given point, project the point onto the extended line containing the segment and clamp the projection point to the segment endpoints.
2) To find the closest points between two lines, set up a system of equations requiring the vector between candidate points to be perpendicular to both lines, yielding a 2x2 linear system whose solution gives the closest points.
3) Closest points on non-parallel lines in 3D are unique, while in 2D lines always intersect unless parallel.
Presentation1
Simultaneous equations are used to solve for two unknowns, such as finding the two constants in two vectors. An example problem finds the values of m and n given vectors for points P, Q, R, and S of a quadrilateral, where the vectors are defined in terms of constants m and n. The solution uses the substitute method for simultaneous equations by substituting equation (2) into equation (1) to find the values of m and n.
Chapter 3 Algebraic Expressions II
This document discusses algebraic expressions and terms. An algebraic term is the product of unknowns and numbers, with the coefficient being the other factors. Like terms have the same unknowns, while unlike terms do not. To multiply algebraic terms, multiply the signs, coefficients, and unknowns. To divide terms, write as a fraction and simplify by cancelling common unknowns in the numerator and denominator. An algebraic expression is a combination of terms using addition and subtraction.
More Related Content
What's hot
Calculus
This lecture discusses inequalities, absolute values, and graphs of solutions to inequalities on a coordinate plane. Key points covered include:
- Inequalities relate values that are different using symbols like < and >. Operations on inequalities follow rules like adding/subtracting a positive number or multiplying/dividing by a negative number requires changing the inequality symbol.
- Absolute value expressions use | | to represent the distance from zero. Operations inside | | follow different rules than normal order of operations.
- Graphing solutions to inequalities involves open/closed circles and drawing lines left/right to represent <, >, ≤, or ≥. Examples are worked through to demonstrate graphing linear and quadratic inequalities.
11.4 power point
The document discusses formulas for finding distance and midpoints between points on a coordinate plane. It provides examples of using the distance formula to calculate distances between points and the Pythagorean theorem to find the length of a hypotenuse. It also demonstrates using the midpoint formula, which takes the average of the x-values and y-values of two points to find the midpoint between them. This is then used along with distance calculations to solve problems involving finding distances or midpoints on a map.
GH 2-6
This document outlines the structure of a mathematical proof and provides an example proof of Theorem 2-7.
The structure of a proof consists of 5 parts: 1) stating the theorem, 2) providing a diagram of given information, 3) listing what is given, 4) listing what is to be proved, 5) providing statements and reasons that lead from the given to the conclusion.
The example proof of Theorem 2-7 states that if two angles are supplements of congruent angles, then the two angles are congruent. The proof uses 5 statements and reasons: 1) stating what is given, 2) defining supplementary angles, 3) substituting equal expressions, 4) using one given fact, 5) applying
Planning a proof
The document outlines the typical structure for proving a theorem, which consists of 5 parts: 1) stating the theorem, 2) illustrating it with a diagram, 3) listing what is given, 4) listing what is to be proved, and 5) a series of statements and reasons leading from the givens to the conclusion. It then provides an example proof of the theorem: "If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent." The proof uses the standard structure and logical reasoning to show that if two pairs of angles are supplementary, then the corresponding angles are congruent.
Mathematical skills II - Rearranging Equations
The document discusses rearranging formulas by making a different variable the subject of the formula. It explains that the process is similar to solving equations, where the same quantity is added or subtracted to both sides, or both sides are multiplied or divided by the same quantity. Examples are provided of rearranging formulas by changing the operations to make a specified variable the subject. The practice problems ask the reader to rearrange several formulas by making the variable listed at the end the subject of each formula.
Reflection of a point about a line & a plane in 2-D & 3-D co-ordinate systems...
Reflection of a point about a line & a plane in 2-D & 3-D co-ordinate systems...Harish Chandra Rajpoot
The document discusses methods for calculating the reflection of points about lines and planes in 2D and 3D coordinate systems. It provides general expressions to find the coordinates of a reflected point given a point and line/plane of reflection. For a point reflected about a line or plane, the midpoint between the original and reflected point must lie on the line/plane and the line connecting the points must be perpendicular/parallel to the line/plane. Using geometry relationships, the author derives formulas to calculate the coordinates of the reflected point and foot of the perpendicular from the original point to the line/plane.Best guess list for cal paper foundation
This document provides a summary of topics that may appear on the Edexcel Foundation Paper 2 exam, including:
1) Finding the lowest common multiple and highest common factor of two numbers.
2) Expressing numbers as products of primes, ordering fractions, decimals and percentages, and converting between them.
3) Solving ratio problems in context and expressing numbers as percentages of other numbers.
Right triangle similarity
This document provides instruction on similarity in right triangles. It begins with examples of identifying similar right triangles and finding geometric means. Geometric mean is defined as the positive square root of the product of two numbers. The document then shows how geometric means can be used to find unknown side lengths in right triangles using proportions. Examples demonstrate applications to measurement problems. The lesson concludes with a quiz reviewing key concepts like writing similarity statements and using proportions with geometric means to solve for missing side lengths in right triangles.
Similarity on right triangle
Similarity on a Right Triangle
Right Triangle with altitude
Right Triangle Theorem
30-60-90 Triangle Theorem
Isosceles Right Triangle Theorem
2.6 prove statements about segments and angles
This document contains examples and explanations of geometric proofs involving segments, angles, and properties of equality and congruence. It includes examples proving statements about segments and angles being equal based on given information. One example proves two statements: (1) that the distance between the food court and music store is the same as the distance between the shoe store and bookstore, and (2) that this is true because the music store is halfway between the food court and shoe store. The document provides guidance on writing two-column geometric proofs and practicing proofs involving segment and angle properties.
2.7 prove angle pair relationships
This document contains examples and explanations of angle pair relationships and angle proofs. It begins with definitions of equal angles if they are both right angles, angles formed by perpendicular lines, and congruent segment angles. Several examples of angle proofs are provided using statements and reasons. Guided practice problems ask to find missing angle measures using angle relationships. The document concludes with an exit slip and homework assignment.
Unit 1. day 17 studyguide1
This document contains various sections on geometry topics including working warm-up problems about writing equations involving angle pairs formed by parallel lines cut by a transversal. It also lists common core standards about transformations of geometric figures and using properties of parallel lines cut by transversals to establish facts about angles. Finally, it reminds students that their exam is on Thursday and includes a study guide they should complete without talking.
Geom9point7 97
This document discusses vectors and their components, magnitude, direction, and addition. It provides examples of calculating the magnitude and component form of vectors. It also explains how to add vectors using the parallelogram method by drawing a parallelogram and finding the diagonal vector sum, or the component method by finding the horizontal and vertical components and adding them. Vector addition is important in physics for the law of conservation of momentum.
Roots of polynomials
The Bairstow method and Muller method are techniques for calculating the roots of polynomials.
The Bairstow method uses synthetic division to iteratively calculate the roots of a polynomial by dividing it by quadratic factors (x^2 - rx - s) until the remainder is zero. It obtains better approximations of r and s at each iteration to isolate the roots.
The Muller method fits a parabola through three initial guesses to obtain coefficients a, b, c. It then uses the quadratic formula on the parabola to find the root, and iterates with new guesses until converging on a solution.
Both methods use iterative techniques to refine initial guesses and isolate the roots of polynomials without using
Factoring
This document provides an overview of different factoring techniques for quadratics, including:
1) Looking for patterns in the terms like perfect squares or a variable being cubed to use shortcuts.
2) Using the greatest common factor when there are only two terms or the coefficient of x is greater than 1.
3) Factoring the difference of squares or perfect square trinomials using their respective shortcuts.
4) Completely factoring polynomials using the generic rectangle/diamond method or variations for more complex expressions.
Math unit38 vectors
This document contains 6 presentations on vectors:
1) Identifying equivalent and opposite vectors in a diagram
2) Calculating components of vectors
3) Writing vector expressions in terms of other vectors
4) Illustrating vector addition and subtraction on a grid
5) Expressing vectors in terms of other vectors using properties of midpoints
6) Expressing vectors in terms of other vectors and proving collinearity using properties of parallelograms
Closest point on line segment to point
This document discusses two methods for finding closest points between line segments and points or between two lines:
1) To find the closest point on a line segment to a given point, project the point onto the extended line containing the segment and clamp the projection point to the segment endpoints.
2) To find the closest points between two lines, set up a system of equations requiring the vector between candidate points to be perpendicular to both lines, yielding a 2x2 linear system whose solution gives the closest points.
3) Closest points on non-parallel lines in 3D are unique, while in 2D lines always intersect unless parallel.
Presentation1
Simultaneous equations are used to solve for two unknowns, such as finding the two constants in two vectors. An example problem finds the values of m and n given vectors for points P, Q, R, and S of a quadrilateral, where the vectors are defined in terms of constants m and n. The solution uses the substitute method for simultaneous equations by substituting equation (2) into equation (1) to find the values of m and n.
What's hot (18)
Reflection of a point about a line & a plane in 2-D & 3-D co-ordinate systems...
Reflection of a point about a line & a plane in 2-D & 3-D co-ordinate systems...
Viewers also liked
Chapter 3 Algebraic Expressions II
This document discusses algebraic expressions and terms. An algebraic term is the product of unknowns and numbers, with the coefficient being the other factors. Like terms have the same unknowns, while unlike terms do not. To multiply algebraic terms, multiply the signs, coefficients, and unknowns. To divide terms, write as a fraction and simplify by cancelling common unknowns in the numerator and denominator. An algebraic expression is a combination of terms using addition and subtraction.
NOTE MATH FORM 3 - 11 linear equations
The document discusses solving simultaneous linear equations through addition, subtraction and multiplication methods. It provides examples of finding the value of one variable given the value of the other variable from two simultaneous linear equations. It also includes exercises for the reader to practice finding the value of variables using the different methods.
Homework chapter 2& 3 form 2
A 8 100 C 8 099 and width (n + 3) cm. If the depth is 5 cm,
calculate the volume of the box in cm3.
B 8 101 D 8 098
34 Simplify: 6x + 3y - (4x - 2y) A 45n2 + 135n + 135 C 45n2 + 135n
B 45n2 + 135n D 45n2 + 135n + 15
A 2x + 5y C 10x + y
B 2x + 5y D 10x - y
39 The perimeter of a rectangle is 60 cm. If
Rancangan Pengajaran Tahunan Matematik Ting.2 2016
RPT MATEMATIK TING.2 2016. Edit daripada guru-guru yang kongsi rpt.
Rancangan pengajaran harian matematik ct hapsyah
Rancangan mengajar matematika tentang perimeter bentuk dua dimensi. Pelajaran ini membincangkan perimeter segiempat sama, segiempat tepat dan segitiga. Murid akan dibimbing untuk mengira perimeter bentuk-bentuk tersebut melalui soalan, diskusi dan aktiviti kumpulan.
Mathematics Mid Year Form 2 Paper 2 2010
This document contains a 20 question mathematics test for Form 2 students. It provides the questions along with space for students to show their work. The test covers topics like operations with integers, decimals, exponents, square roots, simplifying algebraic expressions, solving linear equations, and finding the perimeter of a triangle given side lengths. Students are instructed to answer all questions and show their calculations in the spaces provided. The test is timed for 1 hour and 45 minutes.
Rancangan Pengajaran Harian Matematik
This document contains a daily lesson plan for a mathematics class on improper fractions. It includes:
- Details of the class such as subject, date, time, number of students.
- The learning objectives which are to understand and work with improper fractions.
- A three step plan involving introducing the topic through a video, explaining it using cakes, and then applying it through worksheets.
- Teaching aids used such as video clips, cakes, and worksheets.
- Safety instructions and moral/social values of sharing are also covered.
NOTE MATH FORM 3 - ALGEBRAIC
The document provides examples of expanding and factorizing algebraic expressions involving single and double pairs of brackets. It begins with examples of expanding single pairs of brackets by distributing terms. It then shows examples of expanding double pairs of brackets by distributing the first term over the second pair of brackets. The document also covers finding the highest common factor of algebraic terms and factorizing expressions involving addition or subtraction of like terms.
linear equation
1) The document discusses linear equations in two variables, including defining their form as ax + by = c, explaining that they have infinitely many solutions, and noting that their graphs are straight lines.
2) Specific topics covered include finding solutions, drawing graphs, identifying equations for lines parallel to the x-axis and y-axis, and providing examples of writing and solving linear equations.
3) The summary restates the key points about the properties of linear equations in two variables, such as their graphical and algebraic representations.
PT3,SOALAN AKHIR TAHUN MATEMATIK TINGKATAN 2
Dokumen ini berisi soalan-soalan matematika tingkatan 2 yang meliputi topik-topik seperti operasi algebra, persamaan linear, nisbah, geometri dan transformasi geometri. Terdapat 10 soalan dengan berbagai subsoalan yang mencakup berbagai aspek matematika.
Linear Equations
Students learn to define and identify linear equations. They also learn the definition of Standard Form of a linear equation.
Students also learn to graph linear equations using x and y intercepts.
Chapter 1 Directed Numbers
Numbers to the left of zero on a number line are negative, while numbers to the right are positive. Negative numbers multiplied together are positive, while a negative number multiplied by a positive number or vice versa results in a negative product. Division follows similar rules, with a negative number divided by a positive resulting in a negative quotient. When combining operations, calculations in brackets are performed first, followed by multiplication/division from left to right, and finally addition/subtraction from left to right.
Directed numbers
The document introduces how to add and subtract directed numbers on a number line. It explains that if the directed numbers have the same sign, you add their values, and if they have different signs, you subtract. It provides examples of adding 6 + -3 and subtracting -4 - -9, illustrating starting at the first number on the number line and moving the corresponding number of places based on the operation and second number. The answer is found by labeling the final location.
Directed number
This document provides instruction on adding, subtracting, multiplying, and dividing numbers with negative signs. It begins by defining positive and negative numbers on a number line. It then demonstrates how to perform each operation with positive and negative numbers through examples. Rules are provided for multiplying and dividing with signs, such as two positives or a positive and negative multiplying to a positive or negative. The document encourages practicing these concepts through exercises in a textbook. Overall, it teaches the essential rules and mechanics for performing the basic arithmetic operations with positive and negative numbers.
Viewers also liked (20)
Rancangan Pengajaran Tahunan Matematik Ting.2 2016
Rancangan Pengajaran Tahunan Matematik Ting.2 2016
Similar to Linear Equation
Ratio, Rate and Proportion
The document discusses the definition of ratios and how they relate to vectors. It defines a ratio as the comparison between two quantities that have the same unit. An example shows writing the ratio of 3000 m to 1500 m as 2:1. Ratios are related to vectors because vector problems can use ratios to determine unknown quantities, like using the ratio of 4 PS to SQ to find other vector quantities in a triangle diagram. Mastering ratios is important for understanding vectors since vector concepts apply the ratio concept.
Tmp2023 4
The document discusses the definition of ratios and how they relate to vectors. It defines a ratio as the comparison between two quantities with the same unit. An example shows writing the ratio of 3000 m to 1500 m as 2:1. Ratios are then related to vectors through an example triangle problem where ratios of line segments determine vector components. The key point is that understanding ratios is important for mastering vectors since ratio concepts are applied in vector problems.
FORM 3 & FORM 4
This document discusses the relationships between various mathematical concepts and the topic of vectors, specifically addition and subtraction of vectors. It provides examples of how polygons, parallel lines, algebraic expressions, and straight lines all relate to vectors. Polygons and parallel lines can be used to understand vector addition and subtraction through laws like the parallelogram law. Algebraic expressions allow expressing vectors in terms of other vectors. And vectors are straight lines with direction and magnitude, making straight lines a basic concept for understanding vectors.
Form 3 and Form 4
This document discusses the relationships between various geometry and algebra topics and the topic of vectors, specifically addition and subtraction of vectors. Polygons, parallel lines, algebraic expressions, and straight lines are all related to vectors because vectors represent straight lines with direction and magnitude. Examples are given of using concepts like the parallelogram law, triangle law, and algebraic expressions to solve vector addition and subtraction problems.
Topic 3
This document discusses the relationship between algebraic expressions and vectors. Algebraic expressions are used to solve problems involving addition and subtraction of vectors, where vectors are expressed in terms of other vectors. An example problem involves a triangle PQR where points T and S are given on sides RQ and PQ respectively based on ratios relating distances. The vectors P, S, Q, T, and R must be determined in terms of other vectors using algebraic expressions.
Topic 2
The document discusses angles associated with parallel lines and transversals. When two lines are intersected by a transversal, they are parallel if the corresponding angles are equal, the alternate angles are equal, or the sum of the interior angles is 180 degrees. The concept of parallel lines can be used in vector addition and subtraction, where the resultant vector has the same direction and magnitude as the sum of parallel vectors. An example problem involves finding the measure of an angle in a trapezoid where two bases are parallel.
Discrete-Chapter 08 Relations
The document discusses relations and their properties. It begins by defining a relation as a subset of the Cartesian product of two sets. Relations can be represented using ordered pairs in a set or graphically using arrows. Properties of relations such as reflexive, symmetric, and transitive are introduced. Examples are provided to illustrate relations and calculating their properties. The document also discusses n-ary relations, representing relations using matrices, and operations on relations such as selection.
Boolean variables r010
1. In Boolean algebra, a variable represents a logical quantity that can have a value of 1 or 0. Operations like addition and multiplication represent logical OR and AND operations.
2. Karnaugh maps are used to simplify Boolean expressions by grouping variables and eliminating variables that change between adjacent cells. This groups variables to find the minimum logic expression.
3. Hardware description languages like VHDL and Verilog allow digital designs to be described and implemented using code. VHDL uses entities to describe inputs and outputs, and architectures to describe logic, while Verilog uses modules.
Fundamentals of Physics "VECTORS"
3-1 VECTORS AND THEIR COMPONENTS
After reading this module, you should be able to . . .
3.01 Add vectors by drawing them in head-to-tail arrangements, applying the commutative and associative laws.
3.02 Subtract a vector from a second one.
3.03 Calculate the components of a vector on a given coordinate system, showing them in a drawing.
3.04 Given the components of a vector, draw the vector
and determine its magnitude and orientation.
3.05 Convert angle measures between degrees and radians.
3-2 UNIT VECTORS, ADDING VECTORS BY COMPONENTS
After reading this module, you should be able to . . .
3.06 Convert a vector between magnitude-angle and unit vector notations.
3.07 Add and subtract vectors in magnitude-angle notation
and in unit-vector notation.
3.08 Identify that, for a given vector, rotating the coordinate
system about the origin can change the vector’s components but not the vector itself.
etc...
keph103.pdf
1. The chapter discusses motion in a plane and introduces vectors as a way to describe physical quantities that have both magnitude and direction, such as displacement, velocity, and acceleration.
2. It defines scalars as quantities that only have magnitude and vectors as quantities that have both magnitude and direction, following the triangle law of addition.
3. The chapter then covers adding and subtracting vectors graphically by placing them head to tail, as well as multiplying vectors by real numbers, with negative numbers flipping the direction.
Simple Linear Regression explanation.pptx
These slides describe simple linear regression. Very detail explanation on it. I hope students get more benefits about it.
Simple Linear Regression detail explanation.pdf
These slides describe simple linear regression. Very detail explanation on these slides. I hope students get more benefits about it.
Similar to Linear Equation (12)
More from fatine1232002
Appendix 2 baru
This document contains 4 math problems involving vectors: 1) drawing vectors given vector a and b; 2) using the parallelogram law to find the resultant vector of vectors a and b; 3) determining vectors in a triangle given the midpoint and two other vectors; 4) determining vectors in parallelograms given two initial vectors. It provides diagrams and asks to determine various vectors in terms of the given vectors.
Appendix 1 baru
There are two methods for adding vectors: the parallelogram rule and the triangle rule. The parallelogram rule involves translating the vectors tail-to-tail to form the sides of a parallelogram, where the diagonal lines represent the sum vector. The triangle rule places the tail of one vector at the head of the other to form a triangle, where the third side of the triangle is the sum vector. Both methods are equivalent and result in the sum vector having the same magnitude and direction regardless of the order of the original vectors.
Appendix 2
The document describes a diagram with vectors a and b. Vector a has length PQ which is 4a, and vector b has length SP which is 3b. The question asks to determine the vectors a + b, RQ, and PS in terms of a and b using properties of parallelograms and midpoints. It also asks to determine the vectors PQ and PT in another diagram where PQST and PQRS are parallelograms with sides PQ = 9p and PT = 6q.
Appendix 1
There are two methods for adding vectors: the parallelogram rule and the triangle rule. The parallelogram rule involves translating the vectors tail-to-tail to form the sides of a parallelogram, where the diagonal lines represent the sum of the vectors. The triangle rule places the tail of one vector at the head of the other to form the sides of a triangle, where the third side of the triangle is the sum of the vectors. Both methods are equivalent and result in the commutative property of vector addition where the order of the vectors does not matter.
Baru matrices
Matrices and vectors can be added or subtracted if they are in the same order. Any matrix can be multiplied by a constant. A vector can be represented as the sum of its horizontal and vertical components multiplied by unit vectors. Both matrices and vectors can be added if they are in the same order.
Topic 1
The document discusses the straight line concept and its importance to vectors. It defines a straight line as one without any curves. Straight lines can be described algebraically using linear equations and slope-intercept form. Vectors are also straight lines that do not curve, which is one of the main properties of vectors. Understanding straight lines is important for vectors because it provides knowledge of directions that are used in vector addition and subtraction, where positive and negative vectors combine to form a resultant vector along a straight line.
Topic 1 math (3)
This document discusses how polygons relate to vectors. Polygons can be used to add and subtract vectors using parallelogram and triangle laws. Examples show how vectors can be represented by the sides of polygons, allowing vectors to be easily visualized and understood when solving problems. Polygons are therefore closely related to the topic of vectors in physics and mathematics.
Form 5 (variation vs
This document discusses using the concept of joint variant to write an equation relating variables x, y, and z as y=ax+bz when two vectors p and q are parallel. It provides an example problem to find the possible value of x when p and q are parallel vectors.
Baru matrices
Matrices and vectors can be added or subtracted if they are in the same order. Any matrix can be multiplied by a constant. A vector can be represented as the sum of its horizontal and vertical components multiplied by unit vectors. Both matrices and vectors can be added if they are in the same order.
Topic 1
This document discusses how polygons relate to vectors. Polygons can be used to add and subtract vectors using parallelogram and triangle laws. Examples show how vectors can be represented by the sides of polygons, allowing vectors to be easily visualized and manipulated for problems involving addition and subtraction. Polygons thus provide a useful tool for understanding vector operations.
Line and Angles
This document introduces parallel and perpendicular lines. It defines parallel lines as two lines that never intersect and are always the same distance apart. Perpendicular lines intersect at a 90 degree angle. An example problem in vectors requires knowledge of parallel lines to correctly draw a vector parallel to another given vector.
Tmp2023
This document discusses how integers are related to vectors and how understanding integers can help students learn vectors. It explains that integers, like vectors, have positive and negative values that represent direction. Subtracting vectors is the same as adding the negative of the second vector, and changing a vector's direction requires applying a negative sign. Mastering the concept of positive and negative integers will therefore help students grasp the similar concept of direction for vectors.
Concept Map
Vectors are quantities that have both a magnitude and a direction. Vectors can be represented as an arrow with a length representing the magnitude and the direction it points representing the direction. The magnitude of a vector is commonly represented by its length and the direction is represented by the angle it makes with the x-axis.
Vector
When two variables are related linearly, plotting their values results in a straight line. This straight line of best fit can be used to add or subtract linear relationships, whether the lines are parallel or non-parallel. Vectors can be added by treating the linear lines of best fit for each variable as vectors, with the resultant vector determined by the sums of the components.
Vector
When two variables are related linearly, plotting their values results in a straight line. This straight line of best fit can be used to add or subtract linear relationships, whether the lines are parallel or non-parallel. Vectors can be added by treating the linear lines of best fit for each variable as vectors, with the resultant vector representing their sum.
Presentation1
Simultaneous equations are used to find two unknowns when given two equations. In vectors, simultaneous equations are used to find two constants in two vector equations. This example shows finding the constants m and n given two vector equations relating points P, Q, R, S, and u of a quadrilateral. The values of m and n are found by substituting one vector equation into the other.
Form 5 (variation vs
This document discusses using the concept of joint variant to write an equation relating variables x, y, and z as y=ax+bz when two vectors p and q are parallel. It provides an example problem to find the possible value of x when given vectors p and q are parallel.
Presentation1
Simultaneous equations are used to find two unknowns when there are two equations. In vectors, simultaneous equations are used to find two constants in two vector equations. The example problem gives two vector equations involving points P, Q, R, S, and T on a quadrilateral, with constants m and n, and asks to find the values of m and n by substituting one equation into the other.
FORM 5: Linear Law
When two variables are related linearly, plotting their values results in a straight line. This straight line of best fit can be used to add or subtract linear relationships, whether the lines are parallel or non-parallel. Vectors can be added by treating the linear lines of best fit for each variable as vectors, with the resultant vector representing their sum.
Linear Equation
This document discusses the relationship between linear algebraic expressions and vectors. A linear algebraic expression consists of two or more linear algebraic terms combined using addition, subtraction, or both. Vectors can be represented using linear algebraic expressions, as demonstrated by an example involving a triangle where different vectors are written in terms of other vectors a and b. The document notes that algebraic expressions are commonly used in vectors, and this example is just one of several connections between linear equations and vectors.