Solving Multi-Step Equations

Most real-world problems don't give you tidy, one-step equations. You're more likely to see something like \(3(x + 2) - 5 = 16\), which needs several moves before you can isolate the variable. The good news: if you can solve one-step equations, you already have the skills. Multi-step equations are the same idea with more setup — simplify first, then solve.

The Strategy

Here's the general approach for solving multi-step equations:

Step 1: Simplify each side of the equation if needed (distribute, combine like terms).

Step 2: Get all variable terms on one side and all constant terms on the other.

Step 3: Isolate the variable using inverse operations.

Step 4: Check your answer.

Here's how the strategy plays out on real problems.

Starting Simple: Two Operations

Example: Solve \(2x + 5 = 13\).

This equation has two things happening to \(x\): it's being multiplied by 2, and then 5 is being added. Undo those operations in reverse order — subtract first, then divide.

First, subtract 5 from both sides: $$2x + 5 - 5 = 13 - 5$$ $$2x = 8$$

Now divide both sides by 2: $$\frac{2x}{2} = \frac{8}{2}$$ $$x = 4$$

Check: \(2(4) + 5 = 8 + 5 = 13\) ✓

Example: Solve \(\frac{x}{3} - 7 = 2\).

Add 7 to both sides first: $$\frac{x}{3} - 7 + 7 = 2 + 7$$ $$\frac{x}{3} = 9$$

Now multiply both sides by 3: $$3 \cdot \frac{x}{3} = 3 \cdot 9$$ $$x = 27$$

Check: \(\frac{27}{3} - 7 = 9 - 7 = 2\) ✓

Using the Distributive Property

When there are parentheses, you'll need to distribute before you can combine like terms and solve.

Example: Solve \(4(x - 3) = 20\).

Distribute the 4: $$4x - 12 = 20$$

Add 12 to both sides: $$4x = 32$$

Divide by 4: $$x = 8$$

Check: \(4(8 - 3) = 4(5) = 20\) ✓

Example: Solve \(2(3y + 1) - 5 = 11\).

First, distribute the 2: $$6y + 2 - 5 = 11$$

Combine constants on the left side: $$6y - 3 = 11$$

Add 3 to both sides: $$6y = 14$$

Divide by 6: $$y = \frac{14}{6} = \frac{7}{3}$$

You can leave this as an improper fraction or write it as a mixed number: \(2\frac{1}{3}\). Both are correct.

Check: \(2(3 \cdot \frac{7}{3} + 1) - 5 = 2(7 + 1) - 5 = 2(8) - 5 = 16 - 5 = 11\) ✓

Combining Like Terms on One Side

Sometimes you need to combine like terms before you start solving.

Example: Solve \(5x + 3x - 7 = 17\).

Combine the \(x\) terms on the left: $$8x - 7 = 17$$

Add 7 to both sides: $$8x = 24$$

Divide by 8: $$x = 3$$

Check: \(5(3) + 3(3) - 7 = 15 + 9 - 7 = 17\) ✓

Watch Your Signs with Negative Numbers

Negative signs can be tricky, especially when distributing or combining terms.

Example: Solve \(-3(x + 4) = 15\).

Distribute the -3 (remember, both terms get multiplied by -3): $$-3x - 12 = 15$$

Add 12 to both sides: $$-3x = 27$$

Divide by -3: $$x = -9$$

Check: \(-3(-9 + 4) = -3(-5) = 15\) ✓

Notice that dividing 27 by \(-3\) gives \(-9\), not 9. Signs matter.

Example: Solve \(8 - 2x = 14\).

This one looks different because the variable term is subtracted. Subtract 8 from both sides: $$-2x = 6$$

Divide by -2: $$x = -3$$

Check: \(8 - 2(-3) = 8 + 6 = 14\) ✓

A More Complex Example

Here's a problem that uses every step at once.

Example: Solve \(3(2x - 5) + 4x = 25\).

Step 1: Distribute the 3. $$6x - 15 + 4x = 25$$

Step 2: Combine like terms on the left — what do you get? Show answer\(10x - 15 = 25\). Then add 15: \(10x = 40\). Divide by 10: \(x = 4\). Check: \(3(2(4)-5)+4(4)=3(3)+16=25\) ✓

Practice Time

  1. \(3x + 7 = 22\) Show answerSubtract 7: \(3x = 15\), then divide by 3: \(x = 5\)
  2. \(\frac{m}{5} + 3 = 8\) Show answerSubtract 3: \(\frac{m}{5} = 5\), then multiply by 5: \(m = 25\)
  3. \(2(y - 4) = 10\) Show answerDistribute: \(2y - 8 = 10\), add 8: \(2y = 18\), divide by 2: \(y = 9\)
  4. \(5p - 3p + 8 = 20\) Show answerCombine like terms: \(2p + 8 = 20\), subtract 8: \(2p = 12\), divide by 2: \(p = 6\)
  5. \(-4(a + 2) = 12\) Show answerDistribute: \(-4a - 8 = 12\), add 8: \(-4a = 20\), divide by -4: \(a = -5\)
  6. \(6 - 3x = 15\) Show answerSubtract 6: \(-3x = 9\), divide by -3: \(x = -3\)
  7. \(2(3n + 1) - 5 = 9\) Show answerDistribute: \(6n + 2 - 5 = 9\), combine: \(6n - 3 = 9\), add 3: \(6n = 12\), divide by 6: \(n = 2\)
  8. \(4x + 2x - 10 = 26\) Show answerCombine: \(6x - 10 = 26\), add 10: \(6x = 36\), divide by 6: \(x = 6\)

What's Next?

A few traps to watch for as you practice. When you distribute, every term inside the parentheses gets multiplied — including a negative. So \(-2(x - 3) = -2x + 6\), not \(-2x - 6\). Combining like terms before you solve usually saves work: \(5x + 3x = 24\) collapses to \(8x = 24\) before the division step. And in the final step, divide by the coefficient that's actually attached to the variable: \(7x = 35\) means divide by 7, not by anything else in the original equation.

Once multi-step equations feel routine, you're ready for equations with variables on both sides — the same toolkit, plus one extra step to consolidate the variable on a single side first.

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