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.