How to Solve Simultaneous Equations: Substitution vs Elimination

Simultaneous equations are two (or more) equations that must be true at the same time. To solve a pair, either substitute one equation into the other, or eliminate a variable by adding or subtracting the equations. On a graph, the solution is where the two lines cross.

The example we will use

Method 1: Substitution

1. Rearrange one equation to get one variable on its own. From the second: $x = y - 1$.
2. Substitute into the other equation:

3. Put $y = 3$ back into $x = y - 1$: $x = 2$.

Substitution is best when one variable already has a coefficient of 1 or −1.

Method 2: Elimination

1. Make the coefficients of one variable match. Multiply the second equation by 3:

2. Add it to the first equation so $y$ cancels:

3. Substitute back: $2 - y = -1$, so $y = 3$.

Always check

Put $x = 2$, $y = 3$ into both original equations:

• $2(2) + 3(3) = 4 + 9 = 13$ ✓
• $2 - 3 = -1$ ✓

What it means on a graph

Each linear equation is a straight line. The solution $(2, 3)$ is the point where the lines cross. This explains the special cases:

Example with no solution: $x + 2y = 4$ and $2x + 4y = 3$. Doubling the first gives $2x + 4y = 8$, which contradicts $2x + 4y = 3$. The lines are parallel.

Three equations, three unknowns

The same idea extends to three equations in $x$, $y$ and $z$: eliminate one variable to get two equations in two unknowns, then solve those. For

the solution is $x = 1$, $y = 2$, $z = 3$. With three or more equations, writing the coefficients as a matrix and row-reducing (Gauss–Jordan elimination) keeps things organised.

Common mistakes

• Multiplying only one side of an equation when scaling it.
• Losing a minus sign when subtracting equations.
• Stopping after finding one variable.

Practice questions

1. $x + y = 10$ and $x - y = 4$
2. $3x + 2y = 16$ and $x + 2y = 8$
3. $y = 2x + 1$ and $3x + y = 11$

Answers: 1) $x = 7, y = 3$ 2) $x = 4, y = 2$ 3) $x = 2, y = 5$

Check your answers, and see the lines cross, with the simultaneous equations solver. For single equations use the linear equation solver, and for the matrix method try the matrix calculator.

Frequently asked questions

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