Linear Equation in Two Variables LINEAR EQUATION IN TWO VARIABLES CREATED BY- PRANAY RAJPUT CLASS - X B ROLL NO:- 37
Linear Equations Definition of a Linear Equation A linear equation in two variable x is an equation that can be written in the form ax + by + c = 0, where a ,b and c are real numbers and a and b is not equal to 0. An example of a linear equation in x is .
This is the graph of the equation 2x + 3y = 12. (0,4) (6,0) x 2 -2 Equations of the form ax + by = care called linear equations in two variables. The point (0,4) is the y-intercept. The point (6,0) is the x-intercept.
Solution of an Equation in Two Variables Example: Given the equation 2x + 3y = 18, determine if the ordered pair (3, 4) is a solution to the equation. We substitute 3 in for x and 4 in for y. 2(3) + 3 (4) ? 18 6 + 12 ? 18 18 = 18 True. Therefore, the ordered pair (3, 4) is a solution to the equation 2x + 3y = 18.
The Rectangular Coordinate System Each point in the rectangular coordinate system corresponds to an ordered pair of real numbers (x,y). Note the word “ordered” because order matters. The first number in each pair, called the x-coordinate, denotes the distance and direction from the origin along the x-axis. The second number, the y-coordinate, denotes vertical distance and direction along a line parallel to the y-axis or along the y-axis Itself. In plotting points, we move across first (either left or right), and then move either up or down, alwaysstarting from the origin.
The Rectangular Coordinate System In the rectangular coordinate system, the horizontal number line is the x-axis. The vertical number line is the y-axis. The point of intersection of these axes is their zero points, called the origin. The axes divide the plane into 4 quarters, called quadrants.
CARTESIAN PLANE y-axis Quadrant II ( - ,+) Quadrant I (+,+) x- axis origin Quadrant IV (+, - ) Quadrant III ( - , - )
Plotting Points EXAMPLE Plot the points (3,2) and (-2,-4). SOLUTION (3,2) (-2,-4)
The Graph of an Equation The graph of an equation in two variables is the set of points whose coordinates satisfy the equation. An ordered pair of real numbers (x,y) is said to satisfy the equation when substitution of the x and y coordinates into the equation makes it a true statement.
WHAT ARE SOLUTIONS ? Let ax + by +c = O , where a ,b , c are real numbers such that a and b ≠ O. Then, any pair of values of x and y which satisfies the equation ax + by +c = O, is called a solution of it.
Finding Solutions of an Equation Find five solutions to the equation y = 3x + 1. Start by choosing some x values and then computing the corresponding y values. If x = -2, y = 3(-2) + 1 = -5. Ordered pair (-2, -5) If x = -1, y = 3(-1) + 1 = -2. Ordered pair ( -1, -2) If x =0, y = 3(0) + 1 = 1. Ordered pair (0, 1) If x =1, y = 3(1) + 1 =4. Ordered pair (1, 4) If x =2, y = 3(2) + 1 =7. Ordered pair (2, 7)
Graph of the Equation Plot the five ordered pairs to obtain the graph of y = 3x + 1 (2,7) (1,4) (0,1) (-1,-2) (-2,-5)
TYPES OF METHOD:- TO SOLVE A PAIR OF LINEAR EQUATION IN TWO VARIABLE
ELIMINATION METHOD The method of substitution is not preferable if none of the coefficients of x and y are 1 or -1. For example, substitution is not the preferred method for the system below: 2x – 7y = 3 -5x + 3y = 7 A better method is elimination by addition. The following operations can be used to produce equivalent systems: 1. Two equations can be interchanged. 2. An equation can be multiplied by a non-zero constant. 3. An equation can be multiplied by a non-zero constant and then added to another equation.
SUBSTITUTION METHOD:The first step to solve a pair of linear equations by the substitution method is to solve one equation for either of the variables. The choice of equation or variable in a given pair does not affect the solution for the pair of equations.In the next step, we’ll substitute the resultant value of one variable obtained in the other equation and solve for the other variable.In the last step, we can substitute the value obtained of the variable in any one equation to find the value of the second variable
CROSS MULTIPLICATION METHODLet’s consider the general form of a pair of linear equations.To solve this pair of equations for 𝑥 and 𝑦 using cross-multiplication, we’ll arrange the variables and their coefficients, and , and the constants and We can convert non linear equations in to linear equation by a suitable substitution