GCSE Simultaneous equations (linear) Practice Questions

Add the equations to eliminate y

The y-terms have opposite signs, so they cancel when you add. That leaves one equation in x only.

Solve for x

Divide both sides by 2.

Substitute back to find y

Put $x = 6$ into equation (1) and solve the one-step equation that is left. The solution is $x = 6$, $y = 3$.

Substitute into the first equation

Put $x = 5$ and $y = 2$ into $x + y = 7$. The left-hand side comes to 7, which matches the right-hand side.

Substitute into the second equation

Now test the same pair in $x - y = 3$. It gives 3, which matches again.

Conclude

A solution of simultaneous equations must work in BOTH equations. Checking only one is never enough.

Look at the two equations

The left-hand side of (2) is exactly twice the left-hand side of (1).

Scale equation (1)

Multiplying (1) by 2 gives $2x + 2y = 8$.

Compare with equation (2)

The same expression cannot equal both 8 and 10, so no pair of values can satisfy both equations.

Subtract to see the contradiction

Eliminating gives $0 = 2$, which is impossible. That is the algebraic signature of "no solution".

Rearrange both into y = mx + c

Both lines have gradient -1 but different y-intercepts.

Conclude

Parallel lines never cross, so there is no point that lies on both — the simultaneous equations have no solution.

Label the two equations

Equation (1) already gives one letter in terms of the other, which is the signal to use substitution rather than elimination.

Substitute for y in equation (2)

Replace y with the bracket. Equation (2) now contains only x.

Expand the bracket

Multiply each term inside the bracket by 2.

Collect the x-terms

Add the two x-terms together.

Move the number to the other side

Undo the addition or subtraction so the x-term is on its own.

Solve for x

Divide both sides by 11.

Substitute $x = 2$ into equation (1)

Equation (1) is the quickest way to get y once you know x.

State y

That gives $y = 3$.

Check in equation (2)

Equation (2) was rearranged during the working, so checking the pair in the original version confirms nothing went wrong.

State the solution

Both equations are satisfied, so this is the point where the two lines cross.

Test Marta's values in equation (1)

Substituting $x = 2$ and $y = 3$ into 3x + 2y gives 12, not 13, so her pair fails the first equation.

Test Marta's values in equation (2)

It fails the second equation as well: x - y should be 1, but she gets -1. The sign being reversed is a clue that the values have been swapped.

Label the two equations

Numbering the equations makes it easy to say exactly what you are doing to each one.

Decide which variable to eliminate

To eliminate y the two y-coefficients must be the same size. The lowest common multiple of 2 and 1 is 2, so scale each equation up to that.

Keep equation (1) as it is

The y-coefficient here is already the right size, so equation (1) does not need scaling.

Multiply equation (2) by 2

Multiply every term — both sides — by 2. The equation still has the same solutions, but the y-coefficient is now 2 in size.

Add the scaled equations

The y-terms are now the same size and the signs are opposite, so adding makes them cancel. Adding the two equations removes y completely.

Eliminate y

The y-terms cancel, leaving one equation with only x in it.

Solve for x

Divide both sides by 5.

Substitute $x = 3$ into equation (1)

Put the value you have just found back into one of the original equations to find y.

Simplify

Work out the number part first, then the equation has only one unknown left.

Solve for y

Move the number to the other side, then divide by 2.

Check in equation (2)

Substituting both values into the equation you did not use for the back-substitution confirms the solution is right. Always do this check.

State the solution

These values satisfy both equations at once — they are the coordinates of the point where the two lines cross.

Explain the error

Marta found the right pair of numbers but wrote them against the wrong letters. Always state which value belongs to x and which to y, then check both equations.