Teach Me Maths

Simultaneous Equations (Three Unknowns)

How to reduce three linear equations to two, solve fractional answers and model problems with three unknowns.

Eliminate the same variable twice

Label the three equations. Use two different pairs to eliminate the same variable, leaving two equations in the other two unknowns. Solve that pair, then substitute to find the third value.

Keep the working organised: record which equations you multiply, add or subtract. Check the final three values in all three original equations.

Worked example 1

Solve x+y+z=8, x-y+z=4 and 2x+y-z=5

  1. Subtract the second equation from the first: 2y=4

  2. Your turn. Find y

    y=2

  3. Substitution gives x+z=6 and 2x-z=3. Adding gives 3x=9, so x=3, then z=3
  4. Answer: x=3, y=2, z=3. The original left sides become 8, 4 and 5

Worked example 2

Solve 2x+y-z=0, x-2y+3z=15 and 3x+y+2z=17

  1. Call the equations (1), (2), (3). Equation (3) minus (1) gives x+3z=17
  2. Twice (1) plus (2) gives 5x+z=15. Multiply this by 3 and subtract x+3z=17, giving 14x=28

  3. Your turn. Find x

    x=2

  4. Use 5x+z=15 to find z=5. Then 2(2)+y-5=0, so y=1
  5. Answer: x=2, y=1, z=5

Keep exact values during substitution

Fractions can be valid solutions. Keep them exact while substituting, rather than rounding and creating small errors in the other values. If subtraction feels awkward, write the second equation with every sign reversed, then add.

Worked example 3

Solve x+y+z=4, 2x+2y-z=2 and 4x-2y+z=1

  1. Add the first two equations: 3x+3y=6, so x+y=2
  2. Subtract the first equation from the third: 3x-3y=-3, so x-y=-1

  3. Your turn. Add the reduced equations: 2x=1. Find x

    x=\frac12

  4. Then y=2-\frac12=\frac32, and z=4-\frac12-\frac32=2
  5. Answer: x=\frac12, y=\frac32, z=2

Worked example 4

Two pens, one notebook and one folder cost £12. One pen, two notebooks and one folder cost £13. One pen, one notebook and two folders cost £15. Find each item’s price.

  1. Let the prices in pounds be p,n,f. Then 2p+n+f=12, p+2n+f=13 and p+n+2f=15
  2. The second equation minus the first gives n-p=1. The third minus the second gives f-n=2. Thus n=p+1 and f=p+3

  3. Your turn. Substitute into the first equation: 4p+4=12. Find p

    p=2

  4. Answer: pen £2, notebook £3, folder £5. Substituting these prices gives all three totals.

Worked example 5

A quadratic f(x)=ax^2+bx+c has f(0)=-2, f(1)=3 and f(2)=12. Find a, b and c.

  1. Your turn. Use f(0)=-2 to find c

    c=-2

  2. The other values give a+b-2=3 and 4a+2b-2=12. Hence a+b=5 and 2a+b=7
  3. Subtract to find a=2, then b=3
  4. Answer: a=2, b=3, c=-2, so f(x)=2x^2+3x-2

Common mistakes

  • Eliminating a different variable each time. Eliminate the same variable from two pairs of equations.
  • Losing signs when subtracting. Bracket the whole equation being subtracted.
  • Forgetting the right-hand side. When scaling an equation, multiply both sides.
  • Rounding before back-substitution. Keep exact fractions until all variables are found.
  • Checking only one original equation. Substitute the final triple into all three equations.

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

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