Lesson 2.5

Equations with Grouped Numerators

Sometimes the variable is trapped upstairs. When a numerator is grouped by a fraction bar, treat it like a VIP section—you have to clear the bouncer (the denominator) before you can enter.

Introduction

In , the is grouped together. You cannot subtract 2 yet because the whole group is being divided by 5. To solve this, we must undo the division first.

Past Knowledge

You know that a fraction bar acts like parentheses (grouping symbol).

Today's Goal

Solve equations where the entire numerator is divided by a number.

Future Success

This skill is crucial for working with formulas like the Average ().

Key Concepts

Unlock the Numerator

The denominator locks the numerator in place.
Multiply by the denominator to break the lock.

The 5s cancel out, leaving just .

Compare the Difference

Separate Terms

Here, only $x$ is divided.
Subtract 2 first.

Grouped Terms

Here, $x+2$ is divided.
Multiply by 5 first.

Worked Examples

Example 1: The Standard Group

Basic

Solve for :

1

Step 1: Multiply to Unlock

Multiply both sides by 2.

2

Step 2: Solve the Remaining Equation

Subtract 4 from both sides.

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Example 2: Negative Denominator

Intermediate

Solve for :

1

Step 1: Multiply by Negative

Multiply both sides by -4 to clear the denominator.

2

Step 2: Finish Solving

Add 3 to both sides.

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Example 3: Double Negative Trick

Advanced

Solve for :

1

Step 1: Multiply First

Multiply both sides by 3.

2

Step 2: Isolate Negative k

Subtract 10 from both sides.

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3

Step 3: Flip Signs

Divide by -1 or flip signs.

Common Pitfalls

Subtracting Prematurely

In , you cannot subtract 5 first. The 5 is "trapped inside the fraction house." You must clear the denominator first.

Forgetting to Multiply the Other Side

When multiplying by 3 to clear the fraction, don't forget to multiply the 7 by 3 as well! It becomes .

Real-Life Applications

Calculating grades involves averages. If you have tests scores of 85, 90, and an unknown score $x$, and you want an average of 92:

To solve for what you need on the final test ($x$), you must first multiply by 3 to unlock the sum.

Practice Quiz

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