Math Core

In grade 7 you solved equations like $4x + 9 = 37$ , where the variable shows up on only one side. But real questions often put a variable on both sides. When will two savings accounts hold the same amount? When do two phone plans cost the same? Equations like $5x + 3 = 2x + 18$ answer questions like these, and you can solve them with one extra move.

Collect the variable on one side

The balance rule still holds: whatever you do to one side, you do to the other. That includes adding or subtracting a variable term like $2x$ . After all, $2x$ is just a number you don't know yet, so taking it away from both sides keeps the equation balanced.

The plan is to turn a new problem into an old one. Move all the variable terms to one side, and the equation becomes a two-step equation you already know how to solve.

Solving with variables on both sides

Add or subtract a variable term on both sides so the variable appears on only one side.
Add or subtract a number on both sides so the constants are on the other side.
Multiply or divide to get the variable alone.
Check your answer in the original equation.

Worked example: Getting started

Solve $5x + 3 = 2x + 18$ .

There are $x$ 's on both sides. Subtract $2x$ from both sides to collect them on the left.

\begin{aligned} 5x + 3 &= 2x + 18 \\ 5x - 2x + 3 &= 2x - 2x + 18 \\ 3x + 3 &= 18 \\ 3x &= 15 \\ x &= 5 \end{aligned}

Check: Left side: $5(5) + 3 = 28$ . Right side: $2(5) + 18 = 28$ . ✓

Which side should the variable go on?

Either side works, and you'll get the same answer. A handy habit is to move the variable term with the smaller coefficient. That way the variable term that stays behind has a positive coefficient, and you avoid dividing by a negative number.

Worked example: Keep the coefficient positive

Solve $2a + 9 = 6a - 11$ .

The smaller coefficient is $2$ , so subtract $2a$ from both sides. The $a$ 's end up on the right.

\begin{aligned} 2a + 9 &= 6a - 11 \\ 9 &= 4a - 11 \\ 20 &= 4a \\ 5 &= a \end{aligned}

Writing $5 = a$ is fine. It means the same thing as $a = 5$ .

Check: $2(5) + 9 = 19$ and $6(5) - 11 = 19$ . ✓

Common mistake

When you move a term, you are subtracting it (or adding it) on both sides. It doesn't just jump across. In $7n - 4 = 3n + 20$ , a common mistake is to "move" $3n$ and write $10n - 4 = 20$ . Subtracting $3n$ from both sides really gives $4n - 4 = 20$ , so $n = 6$ .

Decimals and fractions

The steps stay the same when the coefficients are decimals or fractions.

Worked example: A decimal equation

Solve $1.5y + 4 = 0.5y + 10$ .

\begin{aligned} 1.5y + 4 &= 0.5y + 10 \\ 1.5y - 0.5y + 4 &= 10 \\ 1y + 4 &= 10 \\ y &= 6 \end{aligned}

Check: $1.5(6) + 4 = 13$ and $0.5(6) + 10 = 13$ . ✓

Tip

With fractions, you can clear them first. Multiply every term on both sides by the least common denominator. For $\dfrac{3}{4}x - 2 = \dfrac{1}{4}x + 5$ , multiply by $4$ to get $3x - 8 = x + 20$ , which has no fractions at all.

Word problems: when are two amounts equal?

Many "variables on both sides" problems compare two situations that each have a starting amount and a rate. Write one expression for each situation, then set them equal.

Worked example: Comparing two gyms

Gym A charges a $30 sign-up fee plus $5 per visit. Gym B charges a $10 sign-up fee plus $9 per visit. After how many visits do the two gyms cost the same?

Let $v$ be the number of visits.

Gym A costs $30 + 5v$ .
Gym B costs $10 + 9v$ .

Set the costs equal and solve:

\begin{aligned} 30 + 5v &= 10 + 9v \\ 30 &= 10 + 4v \\ 20 &= 4v \\ 5 &= v \end{aligned}

After 5 visits the costs match. Check: Gym A costs $30 + 25 = 55$ dollars, and Gym B costs $10 + 45 = 55$ dollars. ✓