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## 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.

### Example 2.

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:

3 * (Eqn 1) --->
 3* (2x + 3y = 8)
--->    6x + 9y = 24
2 * (Eqn 2) --->
 2 * (3x + 2y = 7)
--->    6x + 4y = 14         Both equations now have the same leading coefficient = 6

Step 2: Subtract the second equation from the first.

 -(6x + 9y = 24 -(6x + 4y = 14) 5y = 10

Step 3: Solve this new equation for y.

 y = 10/5 = 2

Step 4: Substitute y = 2 into either Equation 1 or Equation 2 above and solve for x. We'll use Equation 1.

 2x + 3(2) = 8 2x + 6 = 8 Subtract 6 from both sides 2x = 2 Divide both sides by 2 x = 1

Solution: x = 1, y = 2 or (1,2).

Now study some more worked examples:

 Equation 1: x + y = Equation 2: x + y =

Click on the buttons below to see how to solve these equations.

 ( x + y = ) --------> x + y =    --------> Subtract: x + y = ( x + y = ) --------> x + y =   --------> [ x + y = ] y =

y = (2 d.p.)

Substitute y into Equation 1 and solve for x: x + ( ) =

x = (2 d.p.)

### Exercise 2.

Equation 1: x + y =

Equation 2: x + y =

First determine what you will multiply each of the above equations by to get the same leading coefficients:

 Multiply Equation 1 by Multiply Equation 2 by

Subtract:

 x + y = ( x + y = ) y =

Solve for y. Then substitute y in either Equation 1 or Equation 2 and solve for x. Give your solutions for both x and y to two decimal places:

y = to 2 d.p.

x = to 2 d.p.