Solving Multi-Step Equations

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Concept Review

Solving Multi-Step Equations: The Art of Mathematical Cleanup

Imagine your room is messy—clothes scattered everywhere, books mixed with games, everything jumbled together. Before you can find what you're looking for, you need to organize first. Solving multi-step equations works the same way: you have to combine like terms first before you can isolate the variable.

When we see an equation like 3x + 7 + 2x - 3 = 16, we're looking at mathematical "clutter." Just like you'd group all the shirts together and all the books together, we need to group all the x-terms together and all the number terms together.

The Cleanup Process

Let's work through our messy equation step by step:

Starting equation: 3x + 7 + 2x - 3 = 16

Step 1: Group like terms together

(3x + 2x) + (7 - 3) = 16

Step 2: Combine like terms

5x + 4 = 16

Step 3: Now solve the clean equation

5x = 12

x = 12/5 = 2.4

Think of like terms as identical twins—they look the same because they have the same variable part. The terms 3x and 2x are like twins because they both have "x." The numbers 7 and -3 are like twins because they're both just numbers with no variable attached.

⚡ The "Why Clean First?" Insight

Here's what's counterintuitive: many students try to "solve" before combining like terms. But imagine trying to subtract 7 from both sides of our original messy equation. You'd get:

3x + 2x - 3 = 9

Now you still have to combine like terms! You've just made extra work for yourself. Always clean up first—it's the mathematical equivalent of "measure twice, cut once."

The Universal Pattern

Whether you're dealing with 4a - 2a + 9 = 15 or -2y + 8 + 5y - 1 = 28, the process is always the same: identify like terms (the mathematical twins), combine them into single terms, then solve the resulting clean equation. This pattern works whether you have 2 terms or 20 terms.

🔑 Key Takeaway

Just like cleaning your room makes everything easier to find, combining like terms first makes equations easier to solve. Organization isn't just helpful—it's essential. Master this cleanup process, and you'll solve multi-step equations with confidence every time.

Sample questions

1. Solve for x: 3x + 2x - 5 = 20

✓ x = 5? 5x - 5 = 20, 5x = 25, x = 5

○ x = 5

○ x = 3

○ x = 5

Answer: x = 5? 5x - 5 = 20, 5x = 25, x = 5 — Combine like terms: 5x - 5 = 20, add 5: 5x = 25, divide by 5: x = 5.

2. Solve: 4x - x + 7 = 22

○ x = 5

✓ x = 5? 3x + 7 = 22, 3x = 15, x = 5

○ x = 3

○ x = 5

Answer: x = 5? 3x + 7 = 22, 3x = 15, x = 5 — 4x - x = 3x, so 3x + 7 = 22, subtract 7: 3x = 15, divide: x = 5.

3. Find x: 2x + 3x - 4x + 8 = 18

○ x = 10

○ x = 2

○ x = 10

✓ x = 10? x + 8 = 18, x = 10

Answer: x = 10? x + 8 = 18, x = 10 — 2x + 3x - 4x = x, so x + 8 = 18, subtract 8: x = 10.

Skills in this topic

Solve equations by combining like terms first
Solve equations by using the distributive property first
Solve equations with variables on both sides of the equal sign
Identify equations with no solution or infinite solutions
Clear fractions from an equation by multiplying by the LCD

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