Solving simultaneous equations by the elimination method.

In the worked questions shown below we shall be solving a simultaneous equation using the elimination method.

Question 1 on solving simultaneous equation using elimination.

Find the values of x and y that satisfy both equations:

4x + y = 10 (1)

3x + 3y = 12 (2)

Step 1:

Make the numbers before x the same by multiplying equation 1 by 3 and equation 2 by 4.

4x + y = 10 (1) × 3

3x + 3y = 12 (2) × 4

12x + 3y = 30 (3)

12x + 12y = 48 (4)

Step 2:

Now since the numbers before x are the same we can take the two equations away. Subtract equation 3 from equation 4 to avoid getting negative numbers.

(4) – (3)

9y = 18

Step 3:

Now solve this equation to find the value of y.

9y = 18 (÷9)

y = 2

Step 4:

We have now found that y = 2, so you can now put this into the first equation to find the value of x.

4x + y = 10 (1)

4x + 2 = 10 (-2)

4x = 8

x = 2

So the values of x is also equal to 2

Step 5

All you need to do now is check that the two values you have found satisfy the second equation.

This will confirm that our answers are correct.

3x + 3y = 12

3 × 2 + 3 × 2 = 12

6 + 6 = 12

12 = 12

So both sides of the equation are equal, therefore our two values are correct (x=2 and y=2)

Question 2 on solving simultaneous equation using elimination.

Find the values of x and y that satisfy both equations:

5x + 3y = 36 (1)

4x + y = 26 (2)

Step 1:

In this question I will demonstrate the elimination method by making the coefficients of y the same.

You can do this by multiplying equation 2 by 3

5x + 3y = 36 (1)

4x + y = 26 (2) × 3

5x + 3y = 36 (3)

12x + 3y = 78 (4)

Step 2:

Now since the numbers before y are the same we can take the two equations away. Subtract equation 3 from equation 4 to avoid getting negative numbers.

(4) – (3)

7x = 42

Step 3:

Now solve this equation to find the value of x.

7x = 42 (÷7)

x = 6

Step 4:

We have now found that x = 6, so you can now put this into the first equation to find the value of x.

5x + 3y = 36 (1)

5 × 6 + 3y = 36

30 + 3y = 36 (-30)

3y = 6

y = 2

Step 5

All you need to do now is check that the two values you have found satisfy the second equation.

This will confirm that our answers are correct.

4x + y = 26 (2) × 3

4 × 6 + 2 = 26

26 = 26

So both sides of the equation are equal, therefore our two values are correct (x=6 and y=2).

For more help on solving simultaneous equations using elimination then click here.

Or to take a look at some harder simultaneous equations (ones involving negatives) then click here.