1. Introduction

 

This article will cover how to:

  • solve simultaneous linear equations by elimination

solving simultaneous linear equations by elimination

2. Simultaneous Equations

 

Basic Principles

Simultaneous equation questions have two equations, both of which have two unknown variables. The solution of a pair of simultaneous equations is the values of both variables which make both equations work at the same time.

For example, consider the equations: x + y = 8 and x – y = 2.

x = 1, y = 7 is a solution of the first equation but not the second.

x = 4, y = 2 is a solution of the second equation but not the first.

x = 5, y = 3 is a solution of both equations at the same time, so it is the solution of these simultaneous equations.

The simultaneous equations in this section are linear because the variables are not raised to any higher powers e.g x².

In this section the examples will mostly use x and y but the principles are the same for any pair of letters.

Notes

  • There will normally be one solution (a pair of values) to a pair of simultaneous linear equations.
  • Simultaneous equations can be solved using algebraic methods or graphically.
  • You can always check your answer by substituting the answer into both your original equations

 

3. Solving Simultaneous Linear Equations by Elimination

Elimination Method

This example will illustrate the elimination method. The steps are shown below with their respective working illustration.

 

Solve:

2x = 16 – 5y

7x – 3y = 15

 

Step 1: Number the equations and make sure matching letters are above each other in the pair of equations (usually we get the letters on the left-hand side and the numbers on the right-hand side.

(1)     2x + 5y = 16

(2)     7x – 3y = 15

 

Step 2: Get the coefficients of one of the variables to be the same in both equations by multiplying one or both of the equations by numbers (it doesn’t matter if the signs of the coefficients are different).

Multiply equation (1) by 7 and equation (2) by 2:

(1)     14x + 35y = 112

(2)     14x   – 6y = 30

 

Step 3: Eliminate the variable with the matching coefficients by adding or subtracting the two equations (add if the signs on the matching coefficients are different, or subtract if the signs are the same).

Signs on the matching terms are the same,

so subtract equation (2) from equation (1):

                    41y = 82

(always be careful with negative signs)

 

Step 4: Solve the resulting equation which now only has one variable to get the first part of the solution.

Divide by 41 on both sides: y = 2

 

Step 5: Substitute the first half of the solution into one of the original equations and solve to get the other part of the solution.

Substitute y = 2 into equation (1):

(1)     2x + 10 = 16

                  2x = 6

                    x = 3

 

Step 6: Substitute both parts of the solution into the other original equation to check your answer.

Substitute x=3, y = 2 into equation (2):

(2)     21 – 6 = 15

 

Step 7: Write both parts of the solution clearly.

It works, so the solution is:

                    x = 3

                    y = 2