Multi-step equations

Simplify first, then solve

Some equations look long, but each side can be tidied up before you start undoing anything. Two tools do most of the work: distributing and combining like terms. Both come from simple pictures.

Distributing is area

The area of a rectangle is width times height. Make a rectangle $3$ tall and $x + 4$ wide, split into an $x$ piece and a $4$ piece. Tap each piece to find its area.

$x$

$4$

$3$

??

The whole area is $3(x + 4)$, and it's also the two pieces added together, $3x + 12$. So they must be equal:

$3(x + 4) = 3x + 12$

That's distributing: the $3$ multiplies every part inside the parentheses, because it stretches across the whole width. Once the parentheses are gone, it's a two-step equation.

Like terms are like tiles

Build $4x + 7 + 2x - 3$ out of tiles. The tall bars are $x$, the small squares are $1$, and the red ones are negative. Sort them, then cancel what you can.

$4x + 7 + 2x - 3$

xxxx

1111111

xx

−1−1−1

Group like tilesCancel zero pairs

Like terms are tiles of the same shape: $4x$ and $2x$ are both bars, and $7$ and $-3$ are both squares. A bar and a square can't merge, since $x$ and $1$ are different sizes, which is why $6x + 4$ can't be simplified any further. And a square with its negative makes $0$, so $7 - 3$ leaves $4$.

Watch the negative

What is $-2(x - 5)$? It means two copies of $-(x - 5)$, and taking the negative of a group flips every tile inside it, so each copy is $-x + 5$.

$-x + 5 - x + 5$

−x

11111

−x

11111

Group like tiles

So $-2(x - 5) = -2x + 10$. The $-2 \cdot (-5) = +10$ part is where most mistakes happen.

Putting it together

Distribute first, then combine, then solve.

Key ideas

• Distributing is area: the number outside multiplies every part inside.
• A negative outside the parentheses flips the sign of every term inside.
• Like terms are the same kind of tile. Combine them, then solve the two-step equation that's left.