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Solve Linear Equations
With Rationals

Master equations with fractions, decimals, and rational coefficients: distribute, combine like terms, and isolate the variable.

8th Grade Math

What Is a Linear Equation?

A linear equation is an equation where the variable has an exponent of 1. The graph of its solutions forms a straight line. The general form is:

Our goal is always the same: isolate the variable on one side of the equation to find its value.

What Are Rational Numbers?

A rational number is any number that can be written as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \). This includes:

Fractions
\( \frac{3}{4} \), \( -\frac{2}{5} \), \( \frac{7}{8} \)
Decimals
\( 0.25 \), \( -1.6 \), \( 3.75 \)
Integers
\( 5 = \frac{5}{1} \), \( -3 = \frac{-3}{1} \)

Why Rational Coefficients Matter

Real-world problems rarely give us whole numbers. Discounts, rates, recipes, and measurements all involve fractions and decimals. Being comfortable with rational coefficients means you can solve any linear equation you encounter.

RECIPE
\( \frac{3}{4} \) cup per batch
How many batches? DISCOUNT 0.15 off the price What was the price? SPEED 2.5 miles per hour Time to travel 8 mi?

Each scenario translates into a linear equation with rational coefficients: solving them is the skill we will build today.

Your Two Key Tools

Distributive Property
\( a(b + c) = ab + ac \)

Multiply every term inside the parentheses by the factor outside.
Combining Like Terms
\( 3x + 5x = 8x \)

Add or subtract terms that share the same variable and exponent.

Two Approaches to Fractions

Clear Fractions First Work With Fractions
Strategy Multiply every term by the LCD Add/subtract fractions directly
Best when Multiple different denominators Simple fractions, one denominator
Example start \( \frac{x}{3} + \frac{x}{4} = 7 \) → multiply by 12 \( \frac{x}{3} + 2 = 5 \) → subtract 2
Advantage Eliminates all fractions at once Fewer steps for simple cases

How to Solve Step by Step

1
Clear fractions or decimals (optional): multiply every term by the LCD or by a power of 10.
2
Distribute: apply the distributive property to remove parentheses.
3
Combine like terms on each side of the equation.
4
Move variable terms to one side and constants to the other using inverse operations.
5
Isolate the variable: divide (or multiply) both sides by the coefficient of the variable.
6
Check: substitute your answer back into the original equation to verify.

Example 1: Simple Fractions

Step 1: Add 2 to both sides
\( \frac{3}{4}x - 2 + 2 = 7 + 2 \) → \( \frac{3}{4}x = 9 \)
Step 2: Multiply both sides by \( \frac{4}{3} \) (the reciprocal)
\( \frac{4}{3} \cdot \frac{3}{4}x = 9 \cdot \frac{4}{3} \) → \( x = 12 \)
Check
\( \frac{3}{4}(12) - 2 = 9 - 2 = 7 \) ✓

Example 2: Distribute with Fractions

Step 1: Distribute \( \frac{1}{2} \)
\( \frac{1}{2}x + 3 = \frac{3}{4}x - 1 \)
Step 2: Multiply every term by LCD = 4
\( 4 \cdot \frac{1}{2}x + 4 \cdot 3 = 4 \cdot \frac{3}{4}x - 4 \cdot 1 \) → \( 2x + 12 = 3x - 4 \)
Step 3: Move variables left, constants right
\( 2x - 3x = -4 - 12 \) → \( -x = -16 \) → \( x = 16 \)
Check
\( \frac{1}{2}(16 + 6) = \frac{1}{2}(22) = 11 \) and \( \frac{3}{4}(16) - 1 = 12 - 1 = 11 \) ✓

Example 3: Decimals and Like Terms

Step 1: Combine like terms on the left
\( (0.3x + 0.5x) + 1.2 = 4.7 \) → \( 0.8x + 1.2 = 4.7 \)
Step 2: Subtract 1.2 from both sides
\( 0.8x = 4.7 - 1.2 \) → \( 0.8x = 3.5 \)
Step 3: Divide both sides by 0.8
\( x = \frac{3.5}{0.8} = 4.375 \)
Check
\( 0.3(4.375) + 1.2 + 0.5(4.375) = 1.3125 + 1.2 + 2.1875 = 4.7 \) ✓

The Balance Model

Think of an equation as a balance scale: whatever you do to one side, you must do to the other. This keeps the equation balanced and true.

\( \frac{3}{4}x \)
\( -2 \)
\( 7 \)
=
\( \frac{3}{4}x - 2 = 7 \)
Both sides must stay equal

Clearing Fractions with the LCD

When an equation has multiple fractions, multiply every term by the LCD to eliminate all denominators at once.

WITH FRACTIONS
\( \frac{x}{3} + \frac{x}{4} = 7 \)
× 12 (LCD) FRACTION-FREE
\( 4x + 3x = 84 \)
\( 7x = 84 \rightarrow x = 12 \)
LCD of 3 and 4 is 12: multiply every term by 12

Variables on Both Sides

Step 1: Multiply every term by LCD = 5
\( 2x + 15 = 4x - 5 \)
Step 2: Move variable terms to one side
\( 15 + 5 = 4x - 2x \) → \( 20 = 2x \)
Step 3: Divide both sides by 2
\( x = 10 \)

Example: All Three Skills

Step 1: Distribute \( \frac{2}{3} \)
\( \frac{2}{3} \cdot 3x - \frac{2}{3} \cdot 6 + \frac{1}{2}x = 5 \) → \( 2x - 4 + \frac{1}{2}x = 5 \)
Step 2: Combine like terms (LCD = 2 for \( 2x \) and \( \frac{1}{2}x \))
\( \frac{4}{2}x + \frac{1}{2}x - 4 = 5 \) → \( \frac{5}{2}x - 4 = 5 \)
Step 3: Add 4, then multiply by \( \frac{2}{5} \)
\( \frac{5}{2}x = 9 \) → \( x = 9 \cdot \frac{2}{5} = \frac{18}{5} = 3.6 \)

Common Mistakes to Avoid

✕ Partial Distribution
Only multiplying the first term inside parentheses: \( \frac{1}{2}(x+4) \neq \frac{1}{2}x + 4 \). Distribute to every term.
✕ Forgetting a Term with LCD
Multiplying only fractions by the LCD but not the whole-number terms. Every term on both sides gets multiplied.
✕ Sign Errors
Losing negative signs when distributing or moving terms: \( -(x - 3) = -x + 3 \), not \( -x - 3 \).

Summary: Equation Types

Equation Type First Move Key Skill
\( \frac{3}{4}x = 9 \) Multiply by reciprocal \( \frac{4}{3} \) Inverse operations
\( \frac{x}{3} + \frac{x}{4} = 7 \) Multiply all terms by LCD (12) Clearing fractions
\( \frac{1}{2}(x+6) = 10 \) Distribute \( \frac{1}{2} \) Distributive property
\( 0.3x + 0.5x = 4 \) Combine \( 0.8x = 4 \) Combining like terms
\( \frac{2}{5}x + 3 = \frac{4}{5}x - 1 \) Clear fractions, then collect variables All skills combined

Question 1 of 5

Solve: \( \frac{2}{3}x + 4 = 10 \)

Question 2 of 5

What is the LCD for the equation \( \frac{x}{6} + \frac{x}{4} = 5 \)?

Question 3 of 5

Solve: \( \frac{1}{4}(8x - 12) = 2x + 3 \)

Question 4 of 5

Solve: \( 0.5x + 0.3x - 1.4 = 3.8 \)

Question 5 of 5

Solve: \( \frac{3}{5}(x + 10) = \frac{1}{2}x + 9 \)