Teach Me Maths

Simultaneous Equations (Two Unknowns)

How to solve pairs of equations by elimination or substitution and interpret line–curve intersections.

Both equations must hold for the same pair

A solution is a pair of values that satisfies both equations at once. Elimination is useful for two linear equations. Substitution is especially useful when one equation already gives a variable in terms of the other.

When a quadratic is involved, there may be two pairs, one pair or no real pairs. Keep each output paired with the input that produced it.

Worked example 1

Solve x+y=9 and 2x-y=3

  1. Add the equations to eliminate y: 3x=12

  2. Your turn. Find x

    x=4

  3. Substitute in x+y=9: 4+y=9, so y=5. Check 2(4)-5=3
  4. Answer: x=4, y=5

Worked example 2

Solve 3x+2y=19 and 2x-3y=4

  1. Multiply the first equation by 3: 9x+6y=57. Multiply the second by 2: 4x-6y=8

  2. Your turn. Add to obtain 13x=65. Find x

    x=5

  3. Then 3(5)+2y=19, giving y=2. Check 2(5)-3(2)=4
  4. Answer: x=5, y=2

Worked example 3

Adult tickets cost £7 and child tickets cost £4. A group buys 22 tickets for £118. How many of each ticket are bought?

  1. Let a and c be the numbers of adult and child tickets. Then a+c=22 and 7a+4c=118
  2. Use c=22-a: 7a+4(22-a)=118, so 3a+88=118

  3. Your turn. Solve 3a=30. How many adult tickets?

    a=10

  4. Answer: 10 adult and 12 child tickets. Check 70+48=118 and 10+12=22

Explore how many solutions there are

Try it: compare k=1, k=-\frac14 and k=-1 for y=x+k and y=x^2. A touching line gives one repeated solution.

Substitute the whole expression

If y=2x-1 is substituted into x^2+y^2=13, write x^2+(2x-1)^2=13. The brackets matter: y^2 means squaring the entire expression for y

Worked example 4

Solve y=x+1 and y=x^2-3x-4

  1. Equate the expressions for y: x+1=x^2-3x-4. Rearrange to x^2-4x-5=0

  2. Your turn. Solve (x-5)(x+1)=0

    Give every value, separated by commas

    x=-1 or x=5

  3. Use y=x+1. At x=-1, y=0; at x=5, y=6
  4. Answer: (x,y)=(-1,0) or (5,6)

Worked example 5

Solve y=2x-1 and x^2+y^2=13

  1. Substitute: x^2+(2x-1)^2=13. Expand and rearrange to 5x^2-4x-12=0

  2. Your turn. Solve (5x+6)(x-2)=0

    Give every value, separated by commas

    x=-\frac65 or x=2

  3. At x=2, y=3. At x=-\frac65, y=-\frac{12}5-1=-\frac{17}5
  4. Answer: (2,3) or (-\frac65,-\frac{17}5). Each pair lies on both the line and the circle.

Worked example 6

Solve x+y=7 and xy=10

  1. Use y=7-x. Substitution gives x(7-x)=10, so x^2-7x+10=0
  2. Factorise: (x-2)(x-5)=0, giving x=2 or x=5

  3. Your turn. If x=2, what is the corresponding y?

    When x=2, y=5

  4. Answer: (x,y)=(2,5) or (5,2). These are two different ordered pairs.

Common mistakes

  • Multiplying only some equation terms. Multiply every term on both sides when preparing elimination.
  • Missing signs when subtracting equations. Subtract the whole equation, using brackets.
  • Substituting without brackets. Replace the whole variable with a bracketed expression.
  • Losing a quadratic root. Find every valid root, then substitute each one.
  • Mixing up coordinate pairs. Keep each x value with the y value found from it.

Now try it: Simultaneous Equations (Two Unknowns) practice questions

More on this topic: Simultaneous Equations (Two Unknowns) worksheet with full solutions