Math Core

Real equations are often messier than $2m + 3 = 19$ . They have parentheses, several $x$ terms, or variables on both sides. The trick is to tidy each side first until the equation looks like one you already know how to solve.

The plan

You already have every tool you need. Combining like terms and the distributive property clean up each side. Inverse operations finish the job.

Solving a multi-step equation

Simplify each side. Distribute to remove parentheses, then combine like terms.
Collect the variable on one side. Add or subtract a variable term on both sides.
Solve the two-step equation that's left.
Check in the original equation.

Simplify first

Worked example: Combine like terms

Solve $3x + 4 + 2x = 29$ .

The left side has two $x$ terms. Combine them: $3x + 2x = 5x$ .

\begin{aligned} 3x + 4 + 2x &= 29 \\ 5x + 4 &= 29 \\ 5x &= 25 \\ x &= 5 \end{aligned}

Check: $3(5) + 4 + 2(5) = 15 + 4 + 10 = 29$ . ✓

Worked example: Distribute, then combine

Solve $2(3x - 1) - x = 18$ .

Distribute the $2$ to both terms inside, then combine $6x - x$ .

\begin{aligned} 2(3x - 1) - x &= 18 \\ 6x - 2 - x &= 18 \\ 5x - 2 &= 18 \\ 5x &= 20 \\ x &= 4 \end{aligned}

Check: $2(3 \cdot 4 - 1) - 4 = 2(11) - 4 = 22 - 4 = 18$ . ✓

Common mistake

When you distribute, multiply the outside number by every term inside. A common slip is $2(3x - 1) = 6x - 1$ . The correct result is $6x - 2$ . With a negative outside number, every sign inside changes: $-4(x - 3) = -4x + 12$ .

Variables on both sides

In $7x - 5 = 3x + 19$ , there's an $x$ term on each side. You can't solve until all the $x$ terms are together. Subtract $3x$ from both sides, just as you would subtract a number. The scale stays balanced.

Worked example: Move the variable terms together

Solve $7x - 5 = 3x + 19$ .

\begin{aligned} 7x - 5 - 3x &= 3x + 19 - 3x \\ 4x - 5 &= 19 \\ 4x &= 24 \\ x &= 6 \end{aligned}

Check: Left side: $7(6) - 5 = 37$ . Right side: $3(6) + 19 = 37$ . ✓

Tip

Move the smaller variable term. In $7x - 5 = 3x + 19$ , subtracting $3x$ leaves $4x$ , a positive coefficient. That means fewer negative signs to track.

Worked example: Parentheses on both sides

Solve $4(x - 2) = 2(x + 5)$ .

Distribute on each side, then collect the $x$ terms on the left.

\begin{aligned} 4x - 8 &= 2x + 10 \\ 2x - 8 &= 10 \\ 2x &= 18 \\ x &= 9 \end{aligned}

Check: $4(9 - 2) = 4(7) = 28$ and $2(9 + 5) = 2(14) = 28$ . ✓

When the variable disappears

Sometimes the variable terms cancel completely. What's left tells you how many solutions there are.

$2x + 1 = 2x + 5$ . Subtract $2x$ : $1 = 5$ . That's false, and no value of $x$ can fix it. The equation has no solution.
$3(x + 2) = 3x + 6$ . Distribute: $3x + 6 = 3x + 6$ . Subtract $3x$ : $6 = 6$ . That's always true, so every number is a solution. The equation has infinitely many solutions.

You'll study these special cases more in Algebra 1. For now, don't panic when $x$ vanishes. Just read the statement that's left.