A rectangular vegetable patch is twice as long as it is wide, and 30 m of fencing runs right round it — how wide is it? Because the length is already spelled out in terms of the width, you can slide that straight into the fencing equation and solve for one unknown at a time. That slide is the whole idea of the substitution method.

Two equations, two unknowns. The elimination method adds or subtracts the equations to cancel a variable. Substitution takes a different route: get one variable on its own in one equation, then put that expression in place of the variable in the other. Two unknowns become one.

It shines when an equation already hands you a variable for free. Suppose a shop tells you "a pen costs one pound less than twice a pencil" — that's y = 2x - 1 already — and separately that "three pencils and a pen cost nine pounds", i.e. 3x + y = 9. You know exactly what y is worth in terms of x, so why carry two letters around? Swap it in.

The first equation already has y by itself, so it tells us exactly what y is worth: substitute 2x - 1 for y in the second equation and only x is left.

The recipe is always the same:

1. Isolate one variable in one equation — make it the subject (this is just rearranging the formula).
2. Substitute that expression into the other equation.
3. Solve the single-variable equation you now have.
4. Back-substitute that value to find the second variable.

Pick whichever variable is already alone — or easiest to isolate — so you avoid fractions. Here y is already the subject, so substitution is the natural choice. Doing it: 3x + (2x - 1) = 9, so 5x - 1 = 9, giving 5x = 10 and x = 2. Then back into the first equation: y = 2(2) - 1 = 3. Solution x = 2,\ y = 3.

See it solved

Step through the four moves on our pair — isolate, substitute, solve, back-substitute.

When you have to rearrange first

Often no variable is sitting alone yet. That's fine — spend one line making one the subject, then carry on. Take:

Nothing is isolated, but the -y in the second equation is easy to free: rearrange to y = 3x - 5. Now substitute that whole expression into the first equation:

Back-substitute: y = 3(3) - 5 = 4. Solution x = 3,\ y = 4, and the check holds: 3(3) - 4 = 5. Notice the brackets around (3x - 5) — they made the 2 multiply both terms.

A word problem: how old are they?

Word problems turn a story into two equations. "Amara is three times as old as her brother Ben. In four years' time, their ages will add up to 32." Let a be Amara's age now and b Ben's. The first sentence hands you a variable already alone:

Substitution is the obvious tool — a is the subject of the first equation, so drop 3b into the second:

Ben is 6, so a = 3(6) = 18: Amara is 18. Check the story — in four years they'll be 22 and 10, adding to 32. Once the sentences become equations, it's the same four moves every time.

A curve meets a line — substitution's home turf

Substitution really earns its keep when one equation isn't a straight line. Where does the line y = x + 1 cross the curve y = x^2 - 1? Both already give y on its own, so set the two expressions equal — that's substitution in its purest form:

The single equation became a quadratic, so there are two answers: x = 2 and x = -1. Back-substitute each into y = x + 1: the line meets the curve at (2, 3) and (-1, 0). A line can cut a curve twice, so a quadratic's two roots are the two crossing points — exactly what you'd expect from the picture.

See it explained

Sal Khan works a system by substituting one equation into the other.