Solving One-Step Multiplication/Division Equations

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Concept Review

Solving One-Step Multiplication/Division Equations: The Great Balancing Act

Imagine you're at a carnival game where you need to find how many prizes are hidden in each identical box. You know that 4 boxes contain a total of 20 prizes. How many prizes are in each box? Welcome to the world of solving equations — it's like being a mathematical detective!

When we see an equation like 4x = 20, we're looking at a perfectly balanced scale. The letter x represents our mystery number, and our job is to figure out what it equals. Think of the equals sign as the fulcrum of a seesaw — whatever we do to one side, we must do to the other to keep it balanced.

The Opposite Operation Strategy

Here's the key: to isolate our mystery variable, we use the opposite operation. If our variable is being multiplied, we divide both sides. If it's being divided, we multiply both sides.

Let's solve 4x = 20 step by step:

4x = 20

Since x is multiplied by 4, we divide both sides by 4:

4x ÷ 4 = 20 ÷ 4

x = 5

Check: 4 × 5 = 20 ✓

🔑 The "Undo" Secret

Here's what might surprise you: solving equations is really about undoing what's been done to your variable. It's like working backwards through a recipe!

If multiplied:Divide to undo
If divided:Multiply to undo

Division Equations Work the Same Way

Let's try x ÷ 3 = 7. Since x is being divided by 3, we multiply both sides by 3:

x ÷ 3 = 7

(x ÷ 3) × 3 = 7 × 3

x = 21

Check: 21 ÷ 3 = 7 ✓

This strategy works with any nonnegative rational numbers — whole numbers, fractions, and decimals. Whether you're solving 2.5x = 10 (answer: x = 4) or x ÷ 0.5 = 6 (answer: x = 3), the process stays the same.

🎯 Key Takeaway

Just like figuring out how many prizes are in each carnival box, solving one-step equations is about using opposite operations to "undo" what's hiding your variable. Keep the equation balanced, and you'll always find your answer!

Sample questions

1. To solve the equation 4x = 20, what is the best first step?

○ Subtract 4 from both sides

○ Multiply both sides by 4

✓ Divide both sides by 4

○ Subtract x from both sides

Answer: Divide both sides by 4 — Division is the inverse of multiplication. Dividing by 4 isolates x.

2. If 0.5n = 10, what is the value of n?

○ n = 5

○ n = 20

○ n = 10.5

✓ n = 20? 10 / 0.5 = 20

Answer: n = 20? 10 / 0.5 = 20 — Dividing by 0.5 is the same as multiplying by 2.

3. What happens to the variable x if you divide both sides of 7x = 49 by 7?

✓ It becomes 1x, which is just x

○ It becomes 0

○ It disappears

○ It becomes 7x

Answer: It becomes 1x, which is just x — 7/7 = 1, so the coefficient becomes 1, leaving the variable isolated.

Skills in this topic

Solve equations of the form px = q for cases in which p, q and x are all nonnegative rational numbers
Solve equations of the form x/p = q
Solve one-step equations involving fractional and decimal coefficients
Determine the unknown side length of a geometric figure given the area and one side
Write and solve one-step multiplication or division equations for real-world word problems

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