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Level 7-8 Algebra - Simultaneous Equations
Can you get ten out of ten in this maths quiz?

Level 7-8 Algebra - Simultaneous Equations

Solve two equations at once using substitution and elimination. Learn to find the exact values of x and y in real KS3 contexts.

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Fascinating Fact:

The substitution method replaces one variable with another’s expression, such as substituting y = 3x into 2x + y = 11.

In KS3 Maths, you solve simultaneous equations to find values that make two equations true at the same time. Common methods are substitution and elimination. You’ll check answers by substituting back into both equations.

Key Terms

  • Simultaneous equations: Two or more equations that share the same unknowns and must be true at the same time.
  • Substitution: Replace one variable using an equivalent expression from the other equation, then solve.
  • Elimination: Add or subtract equations (after matching coefficients) to remove one variable and solve the other.

Frequently Asked Questions (Click to see answers)

How do you solve simultaneous equations by substitution?

First, make one variable the subject in an equation, like y = 3x + 1. Substitute this into the other equation and solve for x. Then substitute back to find y.

When should I use the elimination method?

Use elimination when you can match coefficients easily. Multiply one or both equations to line up a variable, add or subtract the equations to eliminate it, then solve and back-substitute.

What if there is no solution or infinitely many solutions?

If the lines are parallel, the equations are inconsistent and there is no solution. If the equations are the same line, there are infinitely many solutions (every point on the line works).

Try These Related Quizzes

1 .
Which of the following equations means exactly the same as 2x + 3y = 13?
4x + 6y = 26
6x + 4y = 26
6x + 6y = 26
6x + 6y = 13
The correct answer is double the original expression. Each term in the expression has been multiplied by two and therefore 2x becomes 4x, 3y becomes 6y and 13 becomes 26
2 .
Which of the following equations means exactly the same as 7x - 5y = 17?
12x - 12y = 51
21x - 15y = 51
21x + 15y = 51
2x - 5y = 51
Here we took the original expression and multiplied each term by three. As long as we multiply each term by the SAME number, the letters in the resulting expression will always have the same value as those in the original expression
3 .
Equation 1 is 2x + 3y = 19. Equation 2 is 3x + 6y = 30. How would we 'balance' one of the terms?
Add the terms in Equation 1 to the terms in Equation 2
Deduct the terms in Equation 1 from Equation 2
Multiply the terms in Equation 1 by 2
Multiply the terms in Equation 2 by 2
We would then have 4x + 6y = 38. The y's in each equation would then match and we would then say that the y's are 'balanced'
4 .
Equation 1 is 8x - 3y = 17. Equation 2 is 4x + y = 21. How would we 'balance' one of the terms?
Add the terms in Equation 1 to the terms in Equation 2
Deduct the terms in Equation 1 from Equation 2
Multiply the terms in Equation 1 by 2
Multiply the terms in Equation 2 by 2
We would then have 8x + 2y = 42. The x's in each equation would be balanced
5 .
We have balanced the y terms in two equations as follows: Equation 1 is 3x + 6y = 30 and Equation 2 is 4x + 6y = 38. What do we now do with them?
Add the terms in Equation 1 to the terms in Equation 2
Take the terms in Equation 1 from the terms in Equation 2
Divide Equation 1 by Equation 2
Multiply Equation 1 by Equation 2
Because the balanced terms (6y) are both positive, we take one from the other to cancel them out and we are then left with a value for x
6 .
We have balanced the x terms in two equations as follows: Equation 1 is 8x - 3y = 17 and Equation 2 is 8x + 2y = 42. What do we now do with them?
Take the terms in Equation 1 from the terms in Equation 2
Add the terms in Equation 1 to the terms in Equation 2
Divide Equation 1 by Equation 2
Multiply Equation 1 by Equation 2
This time we have balanced the x terms and when we take one equation from the other we are left with a value for y. If one value for x had been positive and the other value for x had been negative we would ADD the equations together
7 .
By adding (or subtracting) one equation from another we have concluded that x has a value of 8 in the equation 3x + 6y = 30. What is the value for y?
1
2
3
4
3x = 24
30 - 24 = 6
6 ÷ 6 = 1 therefore y = 1
8 .
By adding (or subtracting) one equation from another we have concluded that y has a value of 5 in the equation 8x - 3y = 17. What is the value for x?
2
4
6
8
3 x 5 = 15
17 + 15 = 32
32 ÷ 8 = 4 therefore x = 4
9 .
What are the values of x and y that can be derived from the following simultaneous equations: 8x - y = 13 and 12x - 3y = 15?
x = 2 and y = 2
x = 2 and y = 3
x = 3 and y = 2
x = 3 and y = 3
Trial and error may help you here!
10 .
What are the values of x and y that can be derived from the following simultaneous equations: 3x + y = 32 and 4x - 2y = 6?
x = 5 and y = 12
x = 6 and y = 10
x = 7 and y = 11
x = 8 and y = 6
That is as difficult as it gets in KS3 maths. If you got all the answers right in this quiz then you have mastered the subject!
You can find more about this topic by visiting BBC Bitesize - Equations

Author:  Frank Evans (Specialist 11 Plus Teacher and Tutor)
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