Multi-Step Equations

Many real-world equations require more than one or two steps to solve. A multi-step equation is any linear equation where you must perform three or more operations — combining like terms, distributing, moving variables across the equals sign — before you arrive at the answer. Once you have the strategy down, even the longest-looking equations become routine.

The General Strategy

Follow this order every time:

1. Simplify each side separately — distribute parentheses and combine like terms.
2. Move variable terms to one side — add or subtract to collect all the variable terms together.
3. Move constant terms to the other side — add or subtract to isolate the variable term.
4. Divide (or multiply) to solve — get the variable alone with a coefficient of 1.

Think of it as tidying up, then isolating.

Combining Like Terms First

Sometimes the same side of an equation has multiple terms with the variable or multiple constant terms. Combine them before doing anything else.

Example 1: $3x + 5 + 2x - 1 = 24$

Step 1 — Combine like terms on the left side:

The variable terms $3x$ and $2x$ combine to $5x$. The constants $5$ and $-1$ combine to $4$.

$5x + 4 = 24$

Step 2 — Subtract 4 from both sides:

$5x = 20$

Step 3 — Divide both sides by 5:

$x = 4$

Answer: $x = 4$

Check: $3(4) + 5 + 2(4) - 1 = 12 + 5 + 8 - 1 = 24$ . Correct.

Example 2: $7 - 2x + 4x + 3 = 18$

Step 1 — Combine like terms on the left:

$2x + 10 = 18$

Step 2 — Subtract 10:

$2x = 8$

Step 3 — Divide by 2:

$x = 4$

Answer: $x = 4$

Using the Distributive Property

When parentheses appear, distribute the factor across every term inside before combining like terms.

Example 3: $4(2x - 3) + 5 = 21$

Step 1 — Distribute the 4:

$8x - 12 + 5 = 21$

Step 2 — Combine constants on the left ($-12 + 5 = -7$):

$8x - 7 = 21$

Step 3 — Add 7 to both sides:

$8x = 28$

Step 4 — Divide by 8:

$x = 3.5$

Answer: $x = 3.5$

Check: $4(2(3.5) - 3) + 5 = 4(7 - 3) + 5 = 4(4) + 5 = 16 + 5 = 21$ . Correct.

Example 4: $-2(x + 6) + 3x = 1$

Pay close attention to distributing the negative.

Step 1 — Distribute $-2$:

$-2x - 12 + 3x = 1$

Step 2 — Combine like terms ($-2x + 3x = x$):

$x - 12 = 1$

Step 3 — Add 12:

$x = 13$

Answer: $x = 13$

Variables on Both Sides — Expanded Treatment

When the variable appears on both sides of the equation, you must move all variable terms to one side and all constants to the other. The key decision is which side to move the variables to — either side works, but choosing the side that keeps the variable coefficient positive avoids sign errors.

Example 5: $6x + 4 = 2x + 20$

The variable appears on both sides. Since $6x$ is larger than $2x$, keep the variable on the left.

Step 1 — Subtract $2x$ from both sides:

$4x + 4 = 20$

Step 2 — Subtract 4 from both sides:

$4x = 16$

Step 3 — Divide by 4:

$x = 4$

Answer: $x = 4$

Check: Left: $6(4) + 4 = 28$. Right: $2(4) + 20 = 28$ . Both sides equal 28.

Example 6: $3x - 7 = 8x + 18$

Here $8x$ is the bigger variable term, so let’s move $3x$ to the right to keep the coefficient positive.

Step 1 — Subtract $3x$ from both sides:

$-7 = 5x + 18$

Step 2 — Subtract 18 from both sides:

$-25 = 5x$

Step 3 — Divide by 5:

$x = -5$

Answer: $x = -5$

Check: Left: $3(-5) - 7 = -15 - 7 = -22$. Right: $8(-5) + 18 = -40 + 18 = -22$ . Both sides equal $-22$.

Example 7: Distributive Property on Both Sides

Solve $5(x + 1) = 3(x - 1) + 12$.

Step 1 — Distribute on both sides:

$5x + 5 = 3x - 3 + 12$

Step 2 — Combine constants on the right ($-3 + 12 = 9$):

$5x + 5 = 3x + 9$

Step 3 — Subtract $3x$:

$2x + 5 = 9$

Step 4 — Subtract 5:

$2x = 4$

Step 5 — Divide by 2:

$x = 2$

Answer: $x = 2$

Special Cases: No Solution and Infinite Solutions

Not every multi-step equation has exactly one answer. There are two special outcomes to watch for.

No Solution (Contradiction)

If the variables cancel and you end up with a false statement, the equation has no solution.

Example: Solve $2(x + 3) = 2x + 10$.

Distribute: $2x + 6 = 2x + 10$

Subtract $2x$: $6 = 10$

This is false — no value of $x$ can make it true. No solution.

Infinite Solutions (Identity)

If the variables cancel and you get a true statement, every real number is a solution.

Example: Solve $3(x - 1) = 3x - 3$.

Distribute: $3x - 3 = 3x - 3$

Subtract $3x$: $-3 = -3$

This is always true. All real numbers are solutions.

Real-World Application: Retail — Comparing Phone Plans

A retail worker is helping a customer compare two phone plans:

• Plan A: $25 per month plus $0.10 per text message
• Plan B: $15 per month plus $0.20 per text message

At how many text messages per month do the two plans cost the same?

Let $t$ represent the number of text messages. Set the costs equal:

$25 + 0.10t = 15 + 0.20t$

Step 1 — Subtract $0.10t$ from both sides:

$25 = 15 + 0.10t$

Step 2 — Subtract 15 from both sides:

$10 = 0.10t$

Step 3 — Divide by 0.10:

$t = 100$

Answer: At 100 text messages, both plans cost the same ($35). Fewer than 100 texts makes Plan B cheaper; more than 100 makes Plan A cheaper.

Real-World Application: Nursing — IV Flow Rate Adjustment

A nurse is recalculating an IV drip. The original order delivers 80 mL per hour. A new medication requires adding 20 mL of solution every 2 hours on top of the base rate. The total volume to infuse in 6 hours is 540 mL.

$80h + \frac{20h}{2} = 540$

Simplify $\frac{20h}{2} = 10h$:

$80h + 10h = 540$

$90h = 540$

$h = 6$

Answer: The 6-hour timeframe is confirmed. The combined rate is 90 mL per hour, which the nurse uses to set the infusion pump correctly.

Common Mistakes to Avoid

1. Distributing to only part of the parentheses. In $4(2x - 3)$, you must multiply 4 by both $2x$ and $-3$: the result is $8x - 12$, not $8x - 3$.
2. Forgetting to distribute the negative sign. $-2(x + 6) = -2x - 12$, not $-2x + 6$.
3. Moving a term without changing its sign. When you subtract $2x$ from both sides, $6x$ becomes $4x$, not $8x$.
4. Stopping too early. If you have $3x = 12$, you still need to divide — the answer is $x = 4$, not $3x = 12$.
5. Not checking special cases. If all the variable terms cancel, determine whether the result is a contradiction (no solution) or an identity (infinite solutions).

Practice Problems

Test your understanding with these problems. Click to reveal each answer.

Key Takeaways

• Multi-step equations require simplifying each side first, then isolating the variable
• Always distribute before combining like terms
• When variables appear on both sides, move them to whichever side keeps the coefficient positive
• Watch for special cases: a false statement means no solution; a true statement means infinite solutions
• Distributing a negative sign is the most common source of errors — check it twice
• Always verify your answer by substituting back into the original equation

Return to Algebra for more topics in this section.