linear equations.pptx
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The document defines and discusses linear equations in one variable. It begins by defining an equation as a statement that two algebraic expressions are equal. It then defines a linear equation in one variable as an equation involving only one variable of the first degree. The document goes on to list properties of equations and methods for solving different types of linear equations in one variable, including using addition/subtraction, multiplication/division, and transposing terms. It also provides examples of solving word problems involving linear equations.
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The document discusses square numbers, square roots, and estimating square roots. It defines a square number as a number that is the product of a whole number multiplied by itself. Square roots are defined as numbers that when multiplied by themselves produce another given number. The document provides examples of calculating square roots of perfect squares by factoring them into smaller perfect square factors. It also describes a method for estimating square roots of non-perfect squares by placing them on a number line between the adjacent perfect squares and interpolating to the nearest tenth.
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- Recall perfect square numbers like 4, 9, 16, 25, etc.
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The correct option is B.
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triangles class9.pptx
1. A triangle is a closed figure formed by three intersecting lines, with three sides, three angles, and three vertices.
2. There are five criteria to determine if two triangles are congruent: side-angle-side (SAS), angle-side-angle (ASA), angle-angle-side (AAS), side-side-side (SSS), and right-angle-hypotenuse-side (RHS).
3. Key properties of triangles include: angles opposite equal sides are equal; sides opposite equal angles are equal; the sum of any two sides is greater than the third side.
Trigonometry class10.pptx
This document provides an overview of trigonometry. It defines trigonometry as dealing with relationships between sides and angles of triangles, particularly right triangles. The origins of trigonometry can be traced back 4000 years to ancient civilizations. Key concepts discussed include right triangles, the Pythagorean theorem, trigonometric ratios like sine, cosine and tangent, and applications of trigonometry in fields like construction, astronomy, and engineering. Examples are provided for using trigonometric functions to solve problems involving heights and distances.
trigonometry.pptx
Trigonometry deals with relationships between the sides and angles of triangles. It originated over 4000 years ago and was used to calculate sundials. Important trigonometric functions include sine, cosine, and tangent, which relate the sides of a right triangle to its angles. Trigonometry has many applications in fields like construction, astronomy, navigation, and engineering.
coordinategeometryclass 10pptx
- Coordinate geometry describes the position of points on a plane using an ordered pair of numbers (x,y). It was proposed by René Descartes to link algebra and geometry.
- To locate a point, we use a pair of perpendicular coordinate axes. The distance from the y-axis is the x-coordinate (abscissa) and the distance from the x-axis is the y-coordinate (ordinate).
- The distance formula uses the differences in x- and y-coordinates of two points (x1,y1) and (x2,y2) to calculate the distance between them as √(x2-x1)2 + (y2-y1)2.
arithmatic progression.pptx
The document defines and provides examples of arithmetic progressions (AP). It states that an AP is a sequence where the difference between consecutive terms is constant. The key characteristics of an AP include:
- The first term is denoted as a
- The common difference is denoted as d
- The nth term is calculated as an = a + (n-1)d
Several examples are provided to illustrate how to determine if a sequence is an AP and how to calculate individual terms and the sum of terms in an AP using the above formulae.
linear equation in two variable.pptx
- The document discusses linear equations in two variables. It defines linear equations and explains that a linear equation in two variables can be written in the form ax + by = c.
- It describes how linear equations in two variables have infinitely many solutions, represented by pairs of x and y values. The graph of a linear equation in two variables is a straight line.
- The document also discusses how equations of lines parallel to the x-axis or y-axis can be represented. The graph of an equation of the form x = a is a line parallel to the y-axis, while an equation of the form y = a graphs as a line parallel to the x-axis.
coordinate geometry.pptx
Coordinate geometry describes the position of points on a plane using an ordered pair of numbers in a Cartesian coordinate system. French mathematician René Descartes developed this system in the 1600s. The system uses two perpendicular axes, the x-axis and y-axis, that intersect at the origin point (0,0). Values to the right of the origin on the x-axis and above the origin on the y-axis are positive, while values to the left and below are negative. Together the axes divide the plane into four quadrants. The document provides examples of finding the abscissa (x-coordinate) and ordinate (y-coordinate) of points, as well as answering questions to test understanding of coordinate geometry concepts.
basic geometric concepts.pptx
This document provides an overview of basic geometric concepts including:
- Geometry deals with properties and relationships of lines, angles, surfaces and solids.
- Key concepts defined include points, line segments, lines, intersecting and parallel lines, rays, polygons, angles, triangles, quadrilaterals, circles, and their various parts.
- Figures like polygons are made up of line segments as sides, and angles are formed at the intersection of two rays originating from the same point.
quadratic equations.pptx
The document defines quadratic equations as polynomial equations of the second degree where the highest exponent is 2. It provides the general form of a quadratic equation as ax2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. There are three main ways to solve quadratic equations: factoring, completing the square, and using the quadratic formula. Factoring involves finding two linear expressions whose product is the quadratic, while completing the square rewrites the equation in the form of a perfect square trinomial. The discriminant, b2 - 4ac, determines the nature of the roots.
square and square root class8.pptx
The document discusses various properties of squares and square roots. It defines a square number as the result of multiplying a number by itself. Some key points made include:
- Numbers ending in 2, 3, 7, or 8 cannot be perfect squares, while numbers ending in 1, 4, 5, 6, or 9 may be squares
- The units digit of a square number depends on the units digit of the original number
- Odd squares are odd numbers and even squares are even numbers
- A number is not a perfect square if it ends in an odd number of zeros
- The Pythagorean theorem relates the squares of the sides of a right triangle
- Square roots can be estimated and found using repeated subtraction
Data Handling class 7.pptx
This document provides information about data handling techniques including bar graphs, pie charts, and pictographs. It defines key terms like data, range, mean, median, and mode. For data, it explains that data is a collection of information gathered for a specific purpose, like test scores or heights. It describes bar graphs as a visual way to represent data with bars of uniform width. Pie charts show the relationship between a whole and its parts, with the circle divided into sectors proportional to the information. Pictographs use pictures to show comparisons. The document also gives formulas for calculating range, mean, and median and provides examples of finding each value from data sets.
linux.pptx
Linux Device Driver controls hardware devices attached to a computer by providing a software interface between the operating system and hardware. It communicates with devices through the computer bus or communication subsystem. A driver allows programs to access hardware functions without knowing specific details by issuing commands to devices when programs make calls, and sending data back to the calling programs from the devices.
Presentation1.pptx
This document outlines the steps involved in 3D reconstruction from images including camera calibration, depth estimation from single images, feature detection and matching between images, point cloud generation from matched features, gridding points into a structured point cloud, and using structure from motion algorithms to reconstruct 3D models from multiple images. Computation can be done on GPUs or CPUs.
Primes Factor Trees.ppt
This document defines key terms related to prime and composite numbers including: product, factor, prime number, composite number, and prime factorization. It provides examples of prime numbers (numbers with exactly two factors) such as 2, 3, 5, 7, and examples of composite numbers (numbers with more than two factors) such as 4, 6, 8, 9. The document explains how to perform prime factorization by writing a composite number as a product of prime factors using a factor tree method in multiple steps. It provides examples of prime factoring various numbers between 4 and 22.
Mexico Economic Crises-1994.pptx
The Mexican peso crisis of 1994 resulted from a sudden devaluation of the peso by 50% in December 1994, triggering a severe recession. Mexico had pursued economic reforms and joined NAFTA but still had a fixed exchange rate, large current account deficit, and government debt denominated in dollars rather than pesos. When political assassinations increased uncertainty, investors fled Mexico rapidly, depleting reserves and forcing further peso devaluations that caused a financial crisis and economic contraction. The U.S. intervened with a $50 billion bailout package that restored stability.
Employee Welfare.pptx
The document discusses employee welfare in India. It defines employee welfare as efforts to improve the lives of workers beyond just wages. It outlines the objectives of welfare, which include improving loyalty, morale, and productivity. It also discusses the agencies that provide welfare, including central/state governments, employers, unions, and other organizations. Common types of statutory and voluntary welfare schemes in India are described.
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linear equations.pptx
- 1. LINEAR EQUATION IN ONE VARIABLE Prepared by: SHARDA CHAUHAN TGT MATHEMATICS
- 2. A statement which states that two algebraic expressions are equal is called an equation. 3x 2y 8 6 x x 9 3X2 5 X 5 The equation involving only one variable in first order is called a linear equation in one variable. 3x50 8y 2 7a153a
- 3. PROPERTIES OF AN EQUATION •If same quantity is added to both sides of the equation, the sums are equal. Thus: x=7 => x+a=7+a •If same quantity is subtracted from both sides of an equation, the differences are equal Thus: x=7 => x-a=7-a •If both the sides of an equation are multiplied by the same quantity, the products are equal. Thus: x=7 => ax=7a •If both the sides of an equation are divided by the same quantity, the quotients are equal. Thus: x=7 => x÷a=7÷a
- 4. TO SOLVE AN EQUATION 1.To solve an equation of the form x+a=b => x+4-4=10-4 (subtracting 4 from both the E.g.: Solve x+4=10 Solution: x+4=10 sides) => x=6 2.To solve an equation of the form x-a=b E.g.: Solve y-6=5 equal. Solution: y-6=5 (adding 6 to both sides) => y-6+6=5+6 => y=11
- 5. 3.TO SOLVE AN EQUATION OF THE FORM AX=B E.G.: SOLVE 3X=9 SOLUTION: 3X=9 4. To solve an equation of the form x/a=b E.g.: Solve = 6 Solution: ×=6×2 3 => x = x 2 => x=12
- 6. SHORT- CUT METHOD (SOLVING AN EQUATION BY TRANSPOSING TERMS) 1.In an equation, an added term is transposed (taken) from one side to the other, it is subtracted. i.e., x+4=10 => x=10-4=6 (4 is transposed) 2. In an equation, a subtracted term is transposed to the other side, it is added. i.e., y-6=5 =>y=5+6=11 3. In an equation, a term in multiplication is transposed to the other side, it is divided. i.e., 3x=12 =>x=12/3=4 4. In an equation a term in division is taken to the other side it is multiplied. i.e => y=6×4=24 (4 is transposed) x (6 is transposed) (3 is transposed)
- 7. TO SOLVE EQUATIONS USING MORE THAN ONE PROPERTY Solve: (1) 3x+8=14 Solution: (transposing 8) 3x=14-8 => 3x=6 => x=6/3 (transposing 3) =>x=2
- 8. (2)2A-3=5 Solution: (transposing 3) (transposing2) 2a=5+3 => 2a=8 => a = 8/2 => a = 4
- 9. (3) 5N/8 =20 Solution: 5n=20×8 => n =204×8/51 => n=4×8=32
- 10. SOL VINGANEQUA TIONWITHV ARIABLEON BOTHTHESIDES Transpose the terms containing the variable, to one side and the constants to the other side . E.g.: (1) Solve 10y-3=7y+9 Solution: 10y-7y = 9+3 (transposing 7y to the left & 3 to the right) => 3y = 12 => y = 12/3 => y = 4
- 11. (2) Solve 2(x-5) + 3(x-2) = 8+7(x-4) Solution: 2x-10+3x-6=8+7x-28 (removing the brackets) => 5x-16 = 7x-20 => 5x-7x = -20+16 => => => -2x = -4 x = -4/-2 x = 2
- 12. SOLVING WORD PROBLEMS •A number increased by 8 equal 15. Find the number? Solution: Let the number be ‘x’ Given, the number increased by 8 equal 15. => x+8 = 15 => x = 15-8 => x = 7
- 13. •A number is decreased by 15 and the new number so obtained is multiplied by 3; the result is 81.Find the number? Solution: Let the number be ‘x’ The number decreased by 15 = x-15 The new number (x-15) multiplied by 3 = 3(x-15) Given 3(x-15) = 81 => 3x-45 = 81 => 3x = 81 + 45 => 3x = 126 => x 3 => x = 42 = 126
- 14. 3) A MAN IS 26 YEARS OLDER THAN HIS SON. AFTER 10 YEARS, HE WILL BE THREE TIMES AS OLD AS HIS SON. FIND THEIR PRESENT AGES? • Solution: let son’s present age= x years Then father’s age = x+26 • After ten years, • Son’s age = x+10 • Father’s age = x+26+10 =x+36 Given, x+36 = 3(x+10) => x+36 =3x+30 => x-3x =30-36 => -2x =-6 => x = =>x=3 •Son’s age = 3 years •Father’s age = 3 + 26=29years