Solving Multi-Step Equations

Leveling Up Your Equation Skills 🚀

In Grade 6 you solved one-step equations like $x + 7 = 12$ or $3x = 15$. Those took a single move to crack.

A multi-step equation needs two or more steps to get the variable alone. They combine the operations you already know, so you'll have to undo them one at a time — in the right order.

Examples of multi-step equations:

• $2x + 5 = 13$ — two steps (subtract, then divide)
• $3(x - 4) = 15$ — three steps (distribute, add, divide)
• $5x - 3 = 2x + 9$ — variables on both sides!

Your goal never changes: get the variable alone on one side of the equal sign.

The Strategy: Unwrap the Present 🎁

Think of an equation like a wrapped gift. The variable is hidden inside, with layers of operations wrapped around it. To reach it, you unwrap in reverse order.

You already know PEMDAS — the order you build an expression:

Parentheses → Exponents → Multiply/Divide → Add/Subtract

To solve, you run that order backwards (reverse PEMDAS):

1. Simplify first — combine like terms and distribute parentheses.
2. Undo addition / subtraction — do the opposite to both sides.
3. Undo multiplication / division — do the opposite to both sides.

The Golden Rule: Whatever you do to one side, you must do to the other side. That keeps the equation balanced, like a scale. ⚖️

A First Worked Example

Solve: $2x + 5 = 13$

Step 1 — Undo the addition. Subtract $5$ from both sides:

$2x + 5 - 5 = 13 - 5 \;\Rightarrow\; 2x = 8$

Step 2 — Undo the multiplication. Divide both sides by $2$:

$\frac{2x}{2} = \frac{8}{2} \;\Rightarrow\; x = 4$

Step 3 — Check. Put $x = 4$ back in: $2(4) + 5 = 8 + 5 = 13$ ✓

Answer: $x = 4$

Notice the order: we undid the $+5$ first, then the $\times 2$. That's the reverse of how you'd build the expression!

Quick Concept Check ✅

Let's make sure the big idea stuck.

Worked Examples: Two-Step Equations ✍️

Example A — Subtraction then division: Solve $3x - 7 = 14$

Step 1 — Add $7$ to both sides (undo the subtraction):

$3x - 7 + 7 = 14 + 7 \;\Rightarrow\; 3x = 21$

Step 2 — Divide both sides by $3$:

$\frac{3x}{3} = \frac{21}{3} \;\Rightarrow\; x = 7$

Check: $3(7) - 7 = 21 - 7 = 14$ ✓ → $x = 7$

Example B — A fraction in the mix: Solve $\dfrac{x}{4} + 3 = 8$

Step 1 — Subtract $3$ from both sides:

$\frac{x}{4} = 5$

Step 2 — Multiply both sides by $4$:

$x = 5 \times 4 = 20$

Check: $\frac{20}{4} + 3 = 5 + 3 = 8$ ✓ → $x = 20$

Worked Example: Use the Distributive Property

When you see parentheses, distribute first to clear them.

Solve: $2(x + 3) = 16$

Step 1 — Distribute the $2$ across the parentheses:

$2 \times x + 2 \times 3 = 16 \;\Rightarrow\; 2x + 6 = 16$

Step 2 — Subtract $6$ from both sides:

$2x = 10$

Step 3 — Divide both sides by $2$:

$x = 5$

Check: $2(5 + 3) = 2(8) = 16$ ✓ → $x = 5$

Watch the sign! With a negative out front, like $-3(x - 4)$, distribute the negative too: $-3x + 12$. Forgetting to flip that sign is the #1 mistake here.

Your Turn ✍️

Solve each equation for $x$. Type just the number (for example, 7 or -3).

• Box 1: $4x + 9 = 25$
• Box 2: $\dfrac{x}{5} - 2 = 3$
• Box 3: $3(x + 2) = 21$

Guided Practice: Choose the Answer 🎯

Work each one carefully on scratch paper, then pick the matching answer.

Fill in the Solving Steps 🧩

Below is the work for solving $3x + 8 = 23$. Choose the option that correctly fills each blank.

Equations in the Real World 🌎

Multi-step equations are perfect for real situations that have a starting fee plus a per-item cost.

Example — Bowling night: A bowling alley charges $3 to rent shoes plus $5 per game. You spent $23 total. How many games did you play?

Set up the equation. Let $g$ = number of games:

$5g + 3 = 23$

The $5g$ is the cost of the games, and the $+3$ is the one-time shoe fee.

Solve it:

• Subtract $3$ from both sides: $5g = 20$
• Divide both sides by $5$: $g = 4$

You played $4$ games. Check: $5(4) + 3 = 20 + 3 = 23$ ✓

The trick to word problems is translating words into an equation. Look for the per-item amount (it multiplies the variable) and the one-time amount (it gets added or subtracted).

Solve the Story Problems 🎟️

Set up an equation, solve it, and type just the number.

• Box 1: A taxi charges a $4 flat fee plus $2 per mile. Your ride cost $18. How many miles was the ride? (Solve $2m + 4 = 18$.)
• Box 2: A gym membership is $20 to join plus $15 per month. You paid $95 total. How many months did you pay for? (Solve $15n + 20 = 95$.)
• Box 3: A pizza is split into slices. After eating $3$ slices, twice the slices left equals $10$. How many slices were left? (Solve $2x = 10$.)

One More Real-World Problem 💰

🏆 Putting It All Together

You can now solve equations that take several steps. Here's the game plan to lock it in:

Two mistakes to avoid:

• Forgetting "both sides." The equation is a balanced scale ⚖️ — change one side, change the other.
• Dropping a negative sign when distributing, like $-3(x - 4) = -3x + 12$. Distribute the negative to every term inside.

Final Challenge: Mixed Practice 🎓

These mix everything together — distributing, two-step solving, and checking. Take your time!