LINEAR-COMBINATION
STEP 1 |
Multiply one or both of the equations by a constant to obtain coefficients that differ only in sign for one of the variables. |
STEP 2 |
Add the revised equations from Step 1. Combining like terms will eliminate one of the variables. Solve for the remaining variable. |
STEP 3 |
Substitute the value obtained in Step 2 into either of the original equations and solve for the other variable |
Example: Solve the linear system using the linear combination method. 2x - 4y = 13 4x-5y = 8 Solution 1.) Multiply the first equation by -2 so that the x-coefficients differ only in sign. -2(2x - 4y = 13) 4x - 5y = 8 -4x + 8y = -26 4x - 5y = 8 2.) Add the revised equation and solve for y. -4x + 8y = -26 4x - 5y = 8 3y = -18 y = -6 3.) Substitute the value of y into one of the original equations. Solve for x. 2x - 4y = 13 2x - 4(-6) = 13 2x + 24 =13 x = -11/2 The solution is (-11/2,-6). You can check this the same way as you did with the substitution method. |
Example: Solve the linear system using the linear combination method. 7x - 12y = -22 -5x + 8y = 14 Solution 1.) Multiply the first equation by 2 and the second equation by 3 so that the coefficients of y differ only in sign. 2(7x - 12y = -22) 3(-5x + 8y = 14) 14x - 24y = -44 -15x + 24y = 42 2.) Add the revised equation and solve for x. 14x - 24y = -44 -15x + 24y = 42 -x = -2 x = 2 3.) Substitute the value of x into one of the original equations. Solve for y. -5x + 8y = 14 -5(2) + 8y = 14 y = 3 The solution is (2,3). You can check this the same way as you did with the substitution method. |