Linear equation in two variables
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This document discusses solving systems of linear equations in two variables. There are three main algebraic methods discussed: 1) elimination by substitution, which involves substituting one variable's expression into the other equation to get an equation with just one variable; 2) elimination by equating coefficients, which involves multiplying equations by constants and subtracting to eliminate one variable; and 3) cross-multiplication, which uses cross-multiplication of fractions to eliminate one variable. An example of using elimination by substitution to solve the system x + 2y = -1 and 2x - 3y = 12 is shown.
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Linear equation in two variables
- 1. LINEAR EQUATION IN TWO VARIABLES Subtitle
- 2. System of equations or simultaneous equations *A pair of linear equations in two variables is said to form a system of simultaneous linear equations *For Example, 2x – 3y + 4 = 0 x + 7y – 1 = 0 *Form a system of two linear equations in variables x and y.
- 3. The general form of a linear equation in two variables x and y is *Ax + by + c = 0 , a =/= 0, b=/=0, where a, b and c being real numbers. *A solution of such an equation is a pair of v-alues, one for x and the other for y, which makes two sides of the equationn equal. * Every linear equation in two variables has infinitely many solutions which can be represented on a certain line.
- 4. For the equation 2x –y = -1 ---(1) 2x +1 = y Y = 2x + 1 3x + 2y = 9 --- (2) 2y = 9 – 3x 9- 3x Y = 2 X 0 2 Y 1 5 X 3 -1 Y 0 6
- 6. ALGEBRAIC METHODS OF SOLVING SIMULTANEOUS LINEAR EQUATIONS The most commonly used algebraic methods of solving simultaneous linear equations in two variables are Method of elimination by substitution Method of elimination by equating the coefficient Method of Cross- multiplication
- 7. ELIMINATION BY SUBSTITUTION STEPS Obtain the two equations. Let the equations be a1x + b1y + c1 = 0 ----------- (i) a2x + b2y + c2 = 0 ----------- (ii) Choose either of the two equations, say (i) and find the value of one variable , say ‘y’ in terms of x Substitute the value of y, obtained in the previous step in equation (ii) to get an equation in x
- 8. ELIMINATION BY SUBSTITUTION Solve the equation obtained in the previous step to get the value of x. Substitute the value of x and get the value of y. Let us take an example x + 2y = -1 ------------------ (i) 2x – 3y = 12 -----------------(ii)
- 9. ELIMINATION METHOD In this method, we eliminate one of the two variables to obtain an equation in one variable which can easily be solved. Putting the value of this variable in any of the given equations, the value of the other variable can be obtained. For example: we want to solve, 3x + 2y = 11 2x + 3y = 4
- 10. Let 3x + 2y = 11 --------- (i) 2x + 3y = 4 ---------(ii) Multiply 3 in equation (i) and 2 in equation (ii) and subtracting eq iv from iii, we get 9x + 6y = 33 ------ (iii) 4x + 6y = 8 ------- (iv) 5x = 25 => x = 5
- 11. putting the value of y in equation (ii) we get, 2x + 3y = 4 2 x 5 + 3y = 4 10 + 3y = 4 3y = 4 – 10 3y = - 6 y = - 2 Hence, x = 5 and y = -2