Practice Maths

Simultaneous Equations: Substitution Method

Key Ideas

Key Terms

Simultaneous equations
Two (or more) equations that must both be true at the same time.
two unknowns
(e.g. x and y) and a unique solution that satisfies both.
substitution method
Rearrange one equation to isolate one variable, then substitute that expression into the other equation.
verify
Your solution by substituting both values back into each original equation.

Steps for the Substitution Method

1. Identify an equation where one variable is isolated (or isolate one).
2. Substitute that expression into the other equation.
3. Solve the resulting single-variable equation.
4. Substitute back to find the second variable.
5. Check both values satisfy both original equations.

Hot Tip Always verify your answer! Substitute both values into both original equations. If either equation is not satisfied, you have made an error somewhere.

Worked Example

Question: Solve simultaneously:   y = 2x + 1   and   x + y = 10

Step 1 — Equation (1) is already solved for y.
y = 2x + 1

Step 2 — Substitute into equation (2).
x + (2x + 1) = 10
3x + 1 = 10

Step 3 — Solve for x.
3x = 9
x = 3

Step 4 — Find y.
y = 2(3) + 1 = 7

Step 5 — Verify.
Eq (1): 7 = 2(3) + 1 = 7 ✓
Eq (2): 3 + 7 = 10 ✓
Solution: x = 3, y = 7

What Are Simultaneous Equations?

Two equations with two unknowns are called simultaneous equations. The solution is a pair of values (x, y) that satisfies both equations at the same time. Graphically, this is the point where two straight lines intersect. For example, if y = x + 2 and y = 3x − 4, the solution is the coordinates of the point where these two lines cross.

The Substitution Method — Step by Step

The substitution method works by replacing one variable with an expression involving the other. Here is the process:

Step 1: Choose one equation and rearrange it to make one variable the subject (e.g. y = ... or x = ...). This step is easiest when a variable already has coefficient 1.

Step 2: Substitute this expression into the other equation, replacing that variable. You now have a single equation with one unknown — solve it.

Step 3: Substitute the value you found back into either original equation to find the other variable.

Step 4: Check: substitute both values into both original equations to confirm they are satisfied.

Example: y = x + 2 and 2x + y = 10. Substitute y = x + 2 into the second: 2x + (x + 2) = 10, so 3x + 2 = 10, giving x = 8/3... Let's use simpler numbers: y = x + 1 and x + y = 7. Substitute: x + (x + 1) = 7, so 2x = 6, x = 3. Then y = 3 + 1 = 4. Solution: (3, 4).

Key tip: Always choose the equation that makes substitution easiest — look for a variable that is already isolated (y = ... or x = ...) or has a coefficient of 1. Substitution works best in this case; when neither variable has coefficient 1, elimination is usually quicker.

When One Variable Is Already Isolated

Substitution is at its most efficient when one equation is already in the form y = (expression) or x = (expression). For example: y = 2x − 3 and 4x − y = 9. The first equation gives y directly, so substitute into the second: 4x − (2x − 3) = 9, which gives 4x − 2x + 3 = 9, so 2x = 6, x = 3. Then y = 2(3) − 3 = 3. Check in equation 2: 4(3) − 3 = 9. ✓

Checking the Solution

After finding both x and y, always verify by substituting the pair into both original equations. This is a reliable way to catch errors without needing to redo the working from scratch. If the check fails for one equation but not the other, the error is most likely in the step where you substituted back to find the second variable.

Mastery Practice

  1. Solve each pair of simultaneous equations using substitution. One variable is already isolated. Fluency

    1. y = x + 3   and   2x + y = 12
    2. y = 3x   and   x + y = 8
    3. y = 4x − 1   and   x + y = 9
    4. y = 2x + 5   and   3x + y = 20
    5. y = 5 − x   and   2x + y = 7
    6. y = x − 4   and   3x − y = 16
    7. x = 2y   and   x + 3y = 10
    8. x = y + 1   and   2x + y = 11
    9. y = 3x − 2   and   2x + y = 13
    10. y = 6 − 2x   and   x + 2y = 7

  2. Rearrange one equation to isolate a variable, then solve by substitution. Show your rearrangement step. Understanding

    1. x + y = 9   and   3x + 2y = 22
    2. 2x + y = 10   and   5x + 3y = 27
    3. x − y = 2   and   4x + y = 23
    4. 3x − y = 4   and   x + 2y = 12
    5. x + 2y = 11   and   2x + 3y = 18
    6. 4x + y = 17   and   2x + 3y = 11

  3. Check whether the given ordered pair is the correct solution to each pair of simultaneous equations. Show full substitution. If incorrect, find the actual solution. Understanding

    1. y = 2x − 1   and   x + y = 8  — claimed solution: x = 3, y = 5
    2. y = x + 4   and   2x + y = 13  — claimed solution: x = 3, y = 7
    3. y = 3x + 1   and   x + y = 9  — claimed solution: x = 2, y = 6

  4. For each problem, define your variables, write two equations, then solve by substitution. Problem Solving

    1. The sum of two numbers is 18. One number is 4 more than the other. Find both numbers.
    2. A pen costs $2 more than a pencil. Together they cost $5.50. Find the cost of each.
    3. There are 24 students in a class. The number of girls is twice the number of boys. How many girls and how many boys are there?
    4. Two friends are saving money. Amara has $30 more than Josh. Together they have $110. How much does each person have?
    5. A rectangle has a perimeter of 36 cm. Its length is 3 cm more than twice its width. Find the dimensions of the rectangle.
    6. At a school fair, adult tickets cost $8 and child tickets cost $5. A family buys 7 tickets in total and pays $44. How many adult and how many child tickets did they buy?

  5. Set up and solve these problems using simultaneous equations and the substitution method. Problem Solving

    1. A cinema sells adult tickets for $12 and student tickets for $8. In one session, 30 tickets are sold for a total of $296. How many adult and student tickets were sold?
    2. A farmer has chickens and cows. There are 20 animals in total and 56 legs altogether. How many chickens and how many cows are there?
    3. Two taps fill a tank. Tap A fills at a rate that is 3 litres per minute faster than Tap B. Together they fill 21 litres per minute. Find the rate of each tap.
    4. The difference between two numbers is 7. When the larger number is doubled and added to the smaller, the result is 40. Find both numbers.

  6. Geometry problems solved with simultaneous equations. Problem Solving

    Strategy. Define variables for the unknown quantities, write two equations from the geometric conditions, then solve by substitution.
    1. Two angles are supplementary (they add to 180°). One angle is 5 times the other. Find both angles.
    2. A rectangle has perimeter 50 cm. Its length is 4 cm more than three times its width. Find the dimensions of the rectangle and its area.
    3. An isosceles triangle has two equal sides of length x cm and a base of y cm. The perimeter is 32 cm. The base is 8 cm shorter than each equal side. Find x and y.

  7. Solve these simultaneous equations. Some involve fractional coefficients or answers. Problem Solving

    1. y = ½x + 3   and   x + y = 9
    2. y = ⅓x − 2   and   2x + y = 11
    3. x + y = 5   and   3x − 2y = 25
    4. 2x − 3y = 1   and   x = 2y − 1

  8. Interpret each pair of simultaneous equations. Attempt to solve by substitution, then decide whether the system has one solution, no solution, or infinitely many solutions. Problem Solving

    Key idea. If substitution gives a contradiction (e.g. 3 = 5), there is no solution (parallel lines). If it gives an identity (e.g. 0 = 0), there are infinitely many solutions (same line).
    1. y = 2x + 1   and   4x − 2y = −2
    2. y = 3x + 4   and   y = 3x − 1
    3. y = −2x + 5   and   2x + y = 5

  9. Rate and mixture problems requiring simultaneous equations. Problem Solving

    1. A cyclist and a runner leave the same point at the same time, travelling in opposite directions. The cyclist travels at 3 times the speed of the runner. After 1 hour they are 48 km apart. Find the speed of each.
    2. A shop sells two types of coffee beans: Type A at $18/kg and Type B at $12/kg. A 5 kg blend costs $78. How many kilograms of each type are in the blend?
    3. Two numbers x and y satisfy: three times x plus twice y equals 29, and x is one more than y. Find x and y.

  10. Extended challenge: multi-step simultaneous equation problems. Problem Solving

    1. A bank account earns simple interest. Account X has $P invested at a rate that earns $80 per year. Account Y has $200 more than Account X invested at a rate that earns $110 per year. The interest rates of both accounts are equal. Let r be the interest rate (as a decimal) and P be the amount in Account X.
      1. Write two equations involving P and r.
      2. Solve for P and r using substitution.
      3. What is the annual interest rate as a percentage?
    2. The graph of the line y = ax + b passes through (2, 7) and (5, 16).
      1. Write two equations in a and b by substituting each point.
      2. Solve for a and b by substitution. Write the equation of the line.
      3. Use the line to predict the value of y when x = 10.
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