The Elimination Method
This method for solving a pair of simultaneous linear equations reduces one equation to one that has only a single variable. Once this has been done, the solution is the same as that for when one line was vertical or parallel. This method is known as the Gaussian elimination method.
Solve the following pair of simultaneous linear equations:Equation 1: 2x + 3y = 8
Equation 2: 3x + 2y = 7
Step 1: Multiply each equation by a suitable number so that the two equations have the same leading coefficient. An easy choice is to multiply Equation 1 by 3, the coefficient of x in Equation 2, and multiply Equation 2 by 2, the x coefficient in Equation 1:
Step 2: Subtract the second equation from the first.
Step 3: Solve this new equation for y.
Step 4: Substitute y = 2 into either Equation 1 or Equation 2 above and solve for x. We'll use Equation 1.
Solution: x = 1, y = 2 or (1,2).
Now study some more worked examples: