Simultaneous Equations — Substitution & Elimination

Simultaneous equations are two or more equations solved together because the same values must satisfy all of them at once. In the usual algebra case, you solve two linear equations in two variables to find one ordered pair $(x, y)$ that makes both equations true.

The two main methods are substitution and elimination. Substitution is usually faster when one variable is already isolated. Elimination is usually faster when a variable will cancel after adding or subtracting the equations.

What a solution to simultaneous equations means

Each equation gives one condition on the same unknowns. A solution works only if it satisfies every condition, not just one of them.

For linear equations, you can also think of the solution as the point where two lines meet. If the lines cross once, there is one solution. If they are parallel, there is no solution. If they are the same line, there are infinitely many solutions.

When to use substitution vs. elimination

Use substitution when a variable is already by itself, or can be isolated without much rearranging. For example, $y = 10 - x$ is easy to place into another equation.

Use elimination when one variable can be cancelled by adding or subtracting the equations. This is especially efficient when the coefficients are already equal or opposites.

Neither method is more correct than the other. The practical question is which one gets you to a clean equation faster.

Worked example: solve a pair of simultaneous equations

Solve

$$x + y = 10$$

and

$$2x - y = 2$$

Method 1: Elimination

This system is friendly for elimination because the $y$ terms are opposites.

Add the equations:

$$(x + y) + (2x - y) = 10 + 2$$

So

$$3x = 12$$

which gives

$$x = 4$$

Now substitute $x = 4$ into $x + y = 10$:

$$4 + y = 10$$

so

$$y = 6$$

The solution is

$$(x, y) = (4, 6)$$

Method 2: Substitution

Start from the first equation:

$$x + y = 10$$

Rearrange it to make one variable the subject:

$$y = 10 - x$$

Substitute that into the second equation:

$$2x - (10 - x) = 2$$

Now simplify:

$$2x - 10 + x = 2$$ $$3x - 10 = 2$$ $$3x = 12$$ $$x = 4$$

Then use $y = 10 - x$:

$$y = 10 - 4 = 6$$

So the solution is again

$$(x, y) = (4, 6)$$

Both methods lead to the same ordered pair because they are solving the same system. The choice is about efficiency, not correctness.

Check the answer in both equations

Check the pair in both original equations:

$$4 + 6 = 10$$

and

$$2(4) - 6 = 8 - 6 = 2$$

Both equations are true, so the solution is correct.

Common mistakes with simultaneous equations

Solving for only one variable

Finding $x$ is not enough if the question asks for the solution to the system. You usually need the full pair.

Losing a negative sign

Sign errors are common in both rearranging and substitution. In the example above, the step $2x - (10 - x)$ must become $2x - 10 + x$, not $2x - 10 - x$.

Choosing a method mechanically

If one variable is already isolated, substitution may be faster. If coefficients already cancel, elimination may be cleaner. Picking the easier route reduces mistakes.

Skipping the check

A wrong answer can still look tidy. Checking both equations is one of the fastest ways to catch a slip.

Where simultaneous equations are used

In school maths, simultaneous equations appear in algebra, graphing, and word problems involving totals, differences, prices, or mixtures. More broadly, they are used whenever two relationships must hold for the same unknown quantities.

The linear case is the usual starting point, but the same idea extends to larger systems and to non-linear equations as well.

Try a similar problem

Solve

$$x + y = 13$$

and

$$3x - y = 7$$

First solve it by elimination. Then solve the same system by substitution and check that both methods give the same ordered pair. If you want another case after that, try changing the constants and see which method becomes quicker.