Gr 8O's 2nd Online Session
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1. The document provides instructions for solving a pair of simultaneous equations graphically. It includes creating tables of values from the equations, plotting the points on a graph, drawing the lines represented by the equations, and finding the point where the lines intersect, which is the solution to the simultaneous equations. 2. An example problem is worked through step-by-step, with the equations y = x + 1 and x + y = 3. Tables of values are completed and the corresponding points are plotted and lines are drawn on the same graph. 3. Another simultaneous equations problem is then posed for students to solve graphically using the same process: if x + y = 1 and x - y = -5,
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Gr 8O's 2nd Online Session (Updated)
1. The document provides instructions on how to solve a pair of simultaneous equations graphically. It includes creating tables of values, plotting points on a graph, drawing lines representing each equation, and finding the point of intersection.
2. An example is worked through step-by-step with equations y = x + 1 and x + y = 3. The solution point is identified as (1,2).
3. A second example is given with equations x + y = 1 and x - y = -5. The reader is instructed to follow the same graphical method to find the solution point.
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This document provides instructions for solving systems of linear inequalities in two variables by graphing. It defines a system of inequalities and explains that the solution is the region where the graphs of the inequalities overlap. A step-by-step process is outlined: 1) graph each inequality individually, 2) shade the appropriate half-plane, 3) the overlapping shaded regions represent the solution. An example system is graphed to demonstrate. Students will evaluate by being assigned a system to graph and answer related questions about the solution region.
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A matrix is an arrangement of items (usually numbers) in rows and columns. Matrices can be added, subtracted, and multiplied following specific rules. The determinant of a matrix is calculated using a calculator function. The inverse of a matrix is found using a calculator function and can be multiplied by the original matrix to yield the identity matrix.
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1) The document discusses the differences between linear equations and inequalities in two variables. Linear equations use the equal sign while inequalities use symbols like <, >, ≤, ≥, ≠.
2) The graph of a linear equation is a single line, while the graph of an inequality shows the shaded region that satisfies the inequality. For < or > the line is broken, and for ≤ or ≥ the line is solid.
3) The document provides steps for graphing a linear inequality in two variables: graph the line by changing the inequality to an equation, use a test point to determine which side to shade, and shade the area where the test point satisfies the inequality.
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1) The document discusses two methods for solving systems of equations: setting the two equations equal to each other (Method 1), or substituting values from one equation into the other (Method 2).
2) An example problem demonstrates Method 1 to find the intersection points of the equations x2+y=4 and y=2x+1. The solutions are x=-3, y=-5 and x=1, y=3.
3) Practice problems provide opportunities to use both Method 1 and Method 2 to solve systems of equations and find their intersection points.
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Gr 8O's 2nd Online Session (Updated)
1. The document provides instructions on how to solve a pair of simultaneous equations graphically. It includes creating tables of values, plotting points on a graph, drawing lines representing each equation, and finding the point of intersection.
2. An example is worked through step-by-step with equations y = x + 1 and x + y = 3. The solution point is identified as (1,2).
3. A second example is given with equations x + y = 1 and x - y = -5. The reader is instructed to follow the same graphical method to find the solution point.
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This document provides instructions for solving systems of linear inequalities in two variables by graphing. It defines a system of inequalities and explains that the solution is the region where the graphs of the inequalities overlap. A step-by-step process is outlined: 1) graph each inequality individually, 2) shade the appropriate half-plane, 3) the overlapping shaded regions represent the solution. An example system is graphed to demonstrate. Students will evaluate by being assigned a system to graph and answer related questions about the solution region.
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A matrix is an arrangement of items (usually numbers) in rows and columns. Matrices can be added, subtracted, and multiplied following specific rules. The determinant of a matrix is calculated using a calculator function. The inverse of a matrix is found using a calculator function and can be multiplied by the original matrix to yield the identity matrix.
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1) The document discusses the differences between linear equations and inequalities in two variables. Linear equations use the equal sign while inequalities use symbols like <, >, ≤, ≥, ≠.
2) The graph of a linear equation is a single line, while the graph of an inequality shows the shaded region that satisfies the inequality. For < or > the line is broken, and for ≤ or ≥ the line is solid.
3) The document provides steps for graphing a linear inequality in two variables: graph the line by changing the inequality to an equation, use a test point to determine which side to shade, and shade the area where the test point satisfies the inequality.
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The document summarizes key information about zeros, roots, or solutions of quadratic functions using a table and graph. It shows a table of x-values and their corresponding f(x) values for a quadratic function. It notes that the points (-2,0) and (2,0) are the x-intercepts, which are also called the zeros, roots, or solutions of the quadratic function, as these are the x-values that make f(x) equal to 0. The document prompts plotting the points on a graph and using a calculator to graph and verify the zeros of the function.
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This document discusses key concepts related to graphing and the rectangular coordinate system. It introduces plotting points in the coordinate plane using ordered pairs (x,y) and defining the four quadrants. It also presents the midpoint formula for finding the midpoint between two points on a line segment by taking the average of the x-coordinates and y-coordinates. An example applies the midpoint formula to find the meeting point for two hikers located at different points on a map.
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1) The document discusses two methods for solving systems of equations: setting the two equations equal to each other (Method 1), or substituting values from one equation into the other (Method 2).
2) An example problem demonstrates Method 1 to find the intersection points of the equations x2+y=4 and y=2x+1. The solutions are x=-3, y=-5 and x=1, y=3.
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The document discusses linear inequalities in two variables. It defines a linear inequality as similar to a linear equation, except the equals sign is replaced with an inequality symbol. Examples of linear inequalities in two variables are provided. The key steps for illustrating and graphing linear inequalities in two variables are outlined, including changing the inequality to an equation, finding the boundary line using x- and y-intercepts, determining whether the boundary line is solid or broken based on the inequality symbol, using a test point to determine which region is shaded, and drawing the final graph showing the shaded solution region.
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This document discusses linear inequalities in two variables and their graphical representations. It can be summarized as:
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This three sentence summary provides the key details about the document:
The document appears to be for a major exam for an electrical engineering course, as it lists the date, time, location, and instructors for the exam and includes 4 problems to be answered. It provides instructions that the exam is closed book, lasts 1.5 hours, and to clearly label sketches. The problems cover topics like probability, conditional probability, networks, and Gaussian random variables.
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The document provides instructions for writing equations of lines given different information. It explains how to write the equation of a line given the slope and a point, or given two points that lie on the line. It also explains how to write the equation of a line given a table of x- and y-values. Finally, it lists homework problems from the textbook for practice writing equations of lines from given information and identifies which problems are due for the next class meeting.
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For more instructional resources, CLICK me here and DON'T FORGET TO SUBSCRIBE!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Reference:
Nivera, G. C. (2013), Grade 7 Mathematics: Pattern and Practicalities. Don Bosco Press Inc. Makati City, Philippines.
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1. The document provides information on linear functions and how to determine if an equation represents a linear function. It discusses using standard form (Ax + By = C) to identify linear equations and determine intercepts.
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This document discusses linear inequalities in two variables and their graphical representations. It can be summarized as:
1) A linear inequality in two variables has infinitely many solutions that can be represented on a coordinate plane as all points on one side of a boundary line.
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2. To solve by factoring, rewrite the equation so one side equals 0, factor the non-zero side, and set each factor equal to 0 to solve.
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A t-chart is used to organize graphing an equation. The document guides the reader through graphing the equation Y=3X+7 by having them: 1) choose three x-values including a negative, positive, and zero; 2) compute the corresponding y-values by plugging the x-values into the equation; 3) plot the ordered pairs (x,y) on a coordinate grid and connect the points.
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This three sentence summary provides the key details about the document:
The document appears to be for a major exam for an electrical engineering course, as it lists the date, time, location, and instructors for the exam and includes 4 problems to be answered. It provides instructions that the exam is closed book, lasts 1.5 hours, and to clearly label sketches. The problems cover topics like probability, conditional probability, networks, and Gaussian random variables.
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The document provides instructions for writing equations of lines given different information. It explains how to write the equation of a line given the slope and a point, or given two points that lie on the line. It also explains how to write the equation of a line given a table of x- and y-values. Finally, it lists homework problems from the textbook for practice writing equations of lines from given information and identifies which problems are due for the next class meeting.
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This document contains a math problem involving classifying different types of numbers and performing calculations with integers, rational numbers, whole numbers, and natural numbers. It asks the reader to:
1) Classify various numbers into the different number categories, identify numbers that don't fit into any category, and draw a map showing the relationships between the categories.
2) Write out the first line of operations and results in English for 4 calculations involving integers, rational numbers, parentheses, and order of operations.
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This document provides information about further maths sessions and matrix algebra. It states that further maths sessions may cover topics students have not been taught or are already experts in, and provides past papers and questions for practice. It also provides instructions on using matrices to represent combined sales data from two shops and how to perform basic matrix operations like addition, subtraction, multiplication by a scalar and between matrices. These include using the correct matrix dimensions and order when performing operations.
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Multiplication and Division of Integers
For more instructional resources, CLICK me here and DON'T FORGET TO SUBSCRIBE!
https://tinyurl.com/y9muob6q
LIKE and FOLLOW me here!
https://tinyurl.com/ycjp8r7u
https://tinyurl.com/ybo27k2u
Reference:
Nivera, G. C. (2013), Grade 7 Mathematics: Pattern and Practicalities. Don Bosco Press Inc. Makati City, Philippines.
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The document discusses differentiation and tangents. It explains that differentiation finds the gradient of a curve at a point and is needed because curves have changing gradients. It provides examples of differentiating simple functions like y=3x^5. Tangents are lines that touch a curve at a single point, with the same gradient as the curve at that point. To find the equation of a tangent, take the derivative and plug in the x-value to get the slope, then use the point to find the y-intercept. Normals are perpendicular to tangents, with a gradient equal to the negative reciprocal of the tangent's gradient.
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The document discusses how to graph quadratic functions by finding the axis of symmetry, vertex, and y-intercept. It explains that the axis of symmetry is located at x = -b/2a, which is the x-coordinate of the vertex. Once the vertex x-coordinate is identified, you can plug it back into the function to find the vertex y-coordinate. The y-intercept is simply the constant term c. It also briefly mentions that for quadratic inequalities, the region will be shaded above or below the parabola depending on whether it is greater than or less than the function.
Introduction to Matrices
A matrix is a rectangular array of numbers arranged in rows and columns. The dimensions of a matrix are written as the number of rows x the number of columns. Each individual entry in the matrix is named by its position, using the matrix name and row and column numbers. Matrices can represent systems of equations or points in a plane. Operations on matrices include addition, multiplication by scalars, and dilation of points represented by matrices.
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- 2. 1. Browse through the slides here. Watch videos on the topic if you have to. 2. Write down the steps for solving in your notebooks. 3. Complete the example given. 4. Post pictures of your table of values (can be done in Microsoft Word), graphs and the solution to the example on Schoology. 5. Complete the Activity Question and post the answer for it on Schoology as well. 6. State an observation you made while completing the activities today (100 points). Click the arrows to move forward or go back.
- 3. Question: Using a graphical method, solve the following pair of simultaneous equations. •y = x + 1 •x + y = 3 Trust me students, you can already guess the answer to that. But every question won’t be so obvious. This is simple. Click the arrow to go back.
- 4. Draw each line that the equation represents. Write down the coordinates for the point at which the two (2) lines intersect/meet.
- 5. Alright, for this question our x values are ranged between – 1 and – 3.
- 6. x - 1 0 1 2 3 Y 0 1 2 Substitute x values from - 1 to 3 in each equation and fill out the table of values. Equation 1: 𝑦 = 𝑥 + 1 1. When x = -1 y = - 1 + 1 y = 0 2. When x = 0 y = 0 + 1 y = 1 3. When x = 1 y = 1 + 1 y = 2 Complete the table by calculating what y would be when x is 2 and when x is 3.
- 7. 1. Draw up a t graph. 2. Number each axis on your graph paper correctly using a scale of 1 unit: 1 cm 3. Plot each point you calculated in the table. For e.g. (-1, 0) (0, 1) (1, 2) 4. Use a ruler to draw a straight line through these points. Label the line as being x + y = 1
- 8. x - 1 0 1 2 3 Y 4 3 2 Substitute x values from - 1 to 3 in each equation and fill out another table of values. Equation 2: x + y = 3 y = 3 - x 1. When x = -1 y = 3 – (-1) y = 3 + 1 y = 4 2. When x = 0 y = 3 – (0) y = 3 3. When x = 1 y = 3 – 1 y = 2 Complete the table by calculating what y would be when x is 2 and when x is 3.
- 9. 1. On the same graph paper as before, plot the coordinates from the 2nd table of values. For e.g. (-1, 4) (0, 3) (1, 2) 2. Use a ruler to draw a straight line through the points. 3. Label this new line as being x + y = 3.
- 10. Where do the two (2) lines intersect/meet?? What is the solution of the simultaneous equations?
- 14. If you add these numbers, the answer is 1 but if you subtract 1 from the other, the answer is -5. x + y = 1 x - y = -5 Using a graphical method, find out what the numbers are. Yes, you are still solving simultaneous equations graphically. 1. Construct your table of values. 2. Label your graph sheet properly. 3. Draw your lines and label them. 4. Write down the point where they intersect. 5. Post all that on Schoology.
- 15. <