Solving Multi-Step Linear Equations

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Concept Review

The Distribution Property: Breaking Down Mathematical Barriers

Imagine you're a construction manager who needs to distribute building materials to different work sites. You have 3 trucks, and each site needs both concrete blocks and steel beams. Instead of making separate trips, you can load each truck with both materials at once. This is exactly how the distribution property works in algebra — it's the most efficient way to "deliver" numbers to multiple terms.

The distribution property states that a(b + c) = ab + ac. When you see a number or variable sitting outside parentheses, it must be multiplied by every term inside those parentheses.

Distribution in Action

Let's solve a real equation step by step: 4(x + 3) = 28

First, we distribute the 4 to both terms inside the parentheses:

4(x) + 4(3) = 28

This gives us: 4x + 12 = 28

Now we can solve normally: subtract 12 from both sides to get 4x = 16, then divide by 4 to find x = 4.

🔍 The Negative Distribution Trap

Here's where many students stumble: distributing negative numbers.

In the equation -2(x - 5) = 10, the negative 2 must be distributed to both terms:

-2(x) + (-2)(-5) = 10
-2x + 10 = 10

Remember: A negative times a negative equals a positive!

When Distribution Gets Complex

Sometimes you'll encounter equations like 3(2x - 4) + 5 = 17. Don't panic! Distribute first: 6x - 12 + 5 = 17. Then combine like terms: 6x - 7 = 17. Add 7 to both sides: 6x = 24. Finally, divide by 6: x = 4.

🔑 Key Insight

Distribution isn't just a rule to memorize — it's a mathematical necessity. You cannot solve equations with parentheses until you eliminate those parentheses through distribution. It's like having to unlock a door before you can enter the room where the solution lives.

Key Takeaway: Just like our construction manager efficiently delivers materials to multiple sites, the distribution property efficiently delivers multiplication to every term inside parentheses. Master this skill, and you'll have the key to unlocking even the most complex multi-step equations. The parentheses aren't barriers — they're just waiting for you to distribute your way through them.

Sample questions

1. Solve for x: 2(x + 3) = 14

✓ x = 4

○ x = 5.5

○ x = 8

○ x = 11

Answer: x = 4 — 2(x + 3) = 14 → 2x + 6 = 14 → 2x = 8 → x = 4.

2. Solve: -3(x - 4) = 21

○ x = 3

✓ x = -3

○ x = -11

○ x = 11

Answer: x = -3 — -3(x - 4) = 21 → -3x + 12 = 21 → -3x = 9 → x = -3.

3. What is the first step to solve 4(2x - 1) = 28?

○ Add 1 to both sides

○ Subtract 1 from both sides

✓ Divide both sides by 4

○ Multiply both sides by 4

Answer: Divide both sides by 4 — You could distribute first, but dividing by 4 first is efficient: (2x - 1) = 7.

Skills in this topic

Solve linear equations requiring the distribution of a positive or negative number
Solve linear equations requiring the combining of like terms on the same side
Solve linear equations with rational number coefficients (fractions and decimals)
Clear denominators in an equation by multiplying by the least common multiple
Verify the solution to a complex equation using substitution

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