Simultaneous Equations

What are simultaneous equations?

Simultaneous equations are two equations that share the same two unknowns — usually x and y — and must be true at the same time. A single equation like x + y = 10 has many solutions: (1, 9), (2, 8), (4, 6) and so on. But pair it with a second equation, and usually only one pair of values works for both.

For example:

x + y = 10 x − y = 4

Only x = 7, y = 3 satisfies both at once: 7 + 3 = 10 and 7 − 3 = 4. That pair is the solution.

This builds directly on solving linear equations; the difference is that now you are juggling two equations and two unknowns together.

Why this matters

Real problems often involve two unknown quantities linked by two conditions. How many adult and child tickets were sold if the total was 50 people and the takings were £320? What are the two numbers whose sum is 15 and difference is 3? Each situation gives you two equations, and solving them simultaneously pins down both unknowns. Scientists, accountants, and engineers do this constantly, often with computers handling dozens of equations at once.

Method 1: Elimination

The idea is to add or subtract the equations so one variable cancels out, leaving a single equation in one unknown.

Worked example: Solve x + y = 10 and x − y = 4.

1. Label them: equation (1) x + y = 10, equation (2) x − y = 4.
2. The y-terms are +y and −y. Add the two equations so they cancel:

(x + y) + (x − y) = 10 + 4 2x = 14

1. Solve: x = 7.
2. Substitute x = 7 back into equation (1): 7 + y = 10, so y = 3.
3. Solution: x = 7, y = 3.
4. Check in equation (2): 7 − 3 = 4. Correct.

Sometimes the coefficients do not match yet, so you scale an equation first.

Worked example: Solve 3x + 2y = 16 and x − 2y = 0.

1. The y-terms are +2y and −2y — they already match in size with opposite signs, so add:

(3x + 2y) + (x − 2y) = 16 + 0 4x = 16, so x = 4.

1. Substitute into x − 2y = 0: 4 − 2y = 0, so 2y = 4 and y = 2.
2. Solution: x = 4, y = 2.

Worked example (with scaling): Solve 2x + 3y = 13 and x + y = 5.

1. Multiply equation (2) by 2 so the x-terms match: 2x + 2y = 10. Call this (2′).
2. Subtract (2′) from (1) to cancel x:

(2x + 3y) − (2x + 2y) = 13 − 10 y = 3.

1. Substitute into x + y = 5: x + 3 = 5, so x = 2.
2. Solution: x = 2, y = 3.

Method 2: Substitution

Here you rearrange one equation to get a variable on its own, then substitute that expression into the other equation.

Worked example: Solve y = 2x and x + y = 9.

1. The first equation already gives y on its own: y = 2x.
2. Substitute 2x in place of y in the second equation: x + 2x = 9.
3. Simplify: 3x = 9, so x = 3.
4. Find y: y = 2x = 2(3) = 6.
5. Solution: x = 3, y = 6.
6. Check: 3 + 6 = 9. Correct.

Worked example: Solve x = y + 1 and 2x + 3y = 12.

1. The first equation gives x = y + 1.
2. Substitute into the second: 2(y + 1) + 3y = 12.
3. Expand: 2y + 2 + 3y = 12.
4. Combine: 5y + 2 = 12, so 5y = 10 and y = 2.
5. Find x: x = y + 1 = 3.
6. Solution: x = 3, y = 2.

Method 3: Graphing

Every linear equation in x and y is a straight line. Plot both lines on the same axes; the point where they cross is the solution, because that point lies on both lines at once. To plot lines confidently, see the coordinate plane.

Worked example: Solve y = x + 1 and y = −x + 5 by graphing.

1. For y = x + 1, the line passes through (0, 1) and (2, 3).
2. For y = −x + 5, the line passes through (0, 5) and (2, 3).
3. Both lines pass through (2, 3) — that is where they cross.
4. Solution: x = 2, y = 3. Substituting confirms it: 3 = 2 + 1 and 3 = −2 + 5.

Graphing gives a great picture but is less precise when the answer is not a whole number, so algebra is usually safer for exact values.

Choosing a method

Situation	Best method	Why
One variable already isolated (e.g. y = 3x − 2)	Substitution	Drop it straight in
Both equations look like ax + by = c	Elimination	Add/subtract to cancel a variable
You want a visual or rough answer	Graphing	See where lines cross
Coefficients already match or are opposites	Elimination	One step cancels immediately

How many solutions?

Most pairs have exactly one solution (lines cross once). But two special cases exist:

• Parallel lines (same slope, different intercept) never meet — no solution.
• Identical lines (the same equation in disguise) overlap completely — infinitely many solutions.

You will spot these when a variable disappears and you are left with something false (like 0 = 5, meaning no solution) or always true (like 0 = 0, meaning infinitely many).

Practice activity

Solve each pair. Choose substitution or elimination.

1. x + y = 7 and x − y = 1
2. y = 3x and x + y = 8
3. 2x + y = 11 and x + y = 7
4. x = 2y and 3x − y = 10

Answers:

1. Add: 2x = 8, so x = 4. Then 4 + y = 7, y = 3. (4, 3).
2. Substitute: x + 3x = 8, 4x = 8, x = 2, y = 6. (2, 6).
3. Subtract the second from the first: x = 4. Then 4 + y = 7, y = 3. (4, 3).
4. Substitute x = 2y: 3(2y) − y = 10, 5y = 10, y = 2, x = 4. (4, 2).

Summary

Simultaneous equations are solved together because their solution must satisfy both equations. Elimination cancels a variable by adding or subtracting (scaling first if needed); substitution replaces one variable using the other equation; graphing finds the crossing point of the two lines. Always substitute your answer back into both original equations to check it is correct.