Solving Multi-Step Equations

Most real-world problems don't give you nice, simple one-step equations. Instead, you get equations like \(3(x + 2) - 5 = 16\) that require multiple steps to solve. The good news? If you can solve one-step equations, you already have all the skills you need. Multi-step equations just require you to 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.

Let's see this in action with actual 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. We need to undo these operations in reverse order.

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 when we divided 27 by -3, we got -9, not 9. The 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

Let's combine everything we've learned.

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

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

Step 2: Combine like terms on the left. $$10x - 15 = 25$$

Step 3: Add 15 to both sides. $$10x = 40$$

Step 4: Divide by 10. $$x = 4$$

Check: \(3(2(4) - 5) + 4(4) = 3(8 - 5) + 16 = 3(3) + 16 = 9 + 16 = 25\) ✓

Practice Time

  1. \(3x + 7 = 22\)
  2. \(\frac{m}{5} + 3 = 8\)
  3. \(2(y - 4) = 10\)
  4. \(5p - 3p + 8 = 20\)
  5. \(-4(a + 2) = 12\)
  6. \(6 - 3x = 15\)
  7. \(2(3n + 1) - 5 = 9\)
  8. \(4x + 2x - 10 = 26\)

Solutions:

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

Avoiding Common Errors

Forgetting to distribute to all terms inside parentheses is probably the most common mistake. When you see \(3(x + 4)\), you must distribute the 3 to both \(x\) and \(4\).

Sign errors with negative numbers trip people up constantly. If you have \(-2(x - 3)\), distributing the -2 gives you \(-2x + 6\), not \(-2x - 6\). A negative times a negative gives a positive.

Not combining like terms before solving makes the problem harder than it needs to be. If you have \(5x + 3x = 24\), combine those \(x\) terms into \(8x\) first.

Dividing by the wrong number happens when people don't pay attention to what coefficient is in front of the variable. If your last step is \(7x = 35\), you divide by 7, not by 5.

Always check your answer by substituting it back into the original equation. If both sides don't equal, you made a mistake somewhere.

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