2.3 Combining the Two Principles

Scott McDaniel

2.3 Combining the Two Principles

Learning Objectives

After completing this section, you will be able to:

Solve multi-step linear equations using both addition/subtraction and multiplication/division principles.
Simplify equations that contain parentheses by applying the distributive property.
Solve equations that contain variables on both sides.
Solve equations involving fractions and decimals (including clearing fractions when appropriate).
Identify whether an equation is conditional, an identity, or a contradiction.

Launch – Putting It All Together

In the last two sections, you learned how to:

Add or subtract to isolate a variable term (Section 2.1).
Multiply or divide to remove coefficients (Section 2.2).

Now you’ll combine both properties to solve multi-step equations. These are equations that require more than one operation and often involve simplification first.

Solving Two-Step Equations

Example 2.3.1

Solve: $2x + 5 = 13$

Solution

Subtract 5 from both sides:

$2x = 8$

Divide both sides by 2:

$x = 4$

✅ Check: $2(4) + 5 = 13$

Example 2.3.2

Solve: $-3x - 7 = 5$

Solution

Add 7 to both sides:

$-3x = 12$

Divide by –3:

$x = -4$

✅ Check: $-3(-4) - 7 = 5$

Try It

Variables on Both Sides

When variables appear on both sides, use addition or subtraction to move one variable term to the other side first.

Example 2.3.3

Solve: $5x - 7 = 2x + 8$

Solution

Subtract $2x$ from both sides:

$3x - 7 = 8$

Add 7:

$3x = 15$

Divide by 3:

$x = 5$

✅ Check: $5(5) - 7 = 2(5) + 8$

Example 2.3.4

Solve: $-4x + 9 = -2x - 3$

Solution

Add $4x$ to both sides:

$9 = 2x - 3$

Add 3:

$12 = 2x$

Divide by 2:

$x = 6$

✅ Check: $-4(6) + 9 = -15$ and $-2(6) - 3 = -15$

Try It

Equations Involving Distribution

Many equations include parentheses that must be simplified before solving. Apply the distributive property first, then combine like terms.

Example 2.3.5

Solve: $3(x + 4) = 27$

Solution

Distribute:

$3x + 12 = 27$

Subtract 12:

$3x = 15$

Divide by 3:

$x = 5$

✅ Check: $3(5 + 4) = 27$

Example 2.3.6

Solve: $-2(x - 5) = 8$

Solution

Distribute:

$-2x + 10 = 8$

Subtract 10:

$-2x = -2$

Divide by –2:

$x = 1$

✅ Check: $-2(1 - 5) = 8$

Example 2.3.7

Solve: $4(x + 2) = 2(x + 6)$

Solution

Distribute:

$4x + 8 = 2x + 12$

Subtract $2x$ from both sides:

$2x + 8 = 12$

Subtract 8:

$2x = 4$

Divide by 2:

$x = 2$

✅ Check: $4(2 + 2) = 2(2 + 6)$

Example 2.3.8

Solve: $3(2x - 5) = 4(x - 1)$

Solution

Distribute:

$6x - 15 = 4x - 4$

Subtract $4x$ from both sides:

$2x - 15 = -4$

Add 15:

$2x = 11$

Divide by 2:

$x = \frac{11}{2} \text{ or } 5.5$

✅ Check:

Left side: $3(2(5.5) - 5) = 3(11 - 5) = 18$

Right side: $4(5.5 - 1) = 4(4.5) = 18$

Example 2.3.9

Solve: $\frac{1}{2}(x + 8) = 6$

Solution

Distribute:

$\frac{1}{2}x + 4 = 6$

Subtract 4:

$\frac{1}{2}x = 2$

Multiply by 2:

$x = 4$

✅ Check: $\frac{1}{2}(4 + 8) = \frac{1}{2}(12) = 6$

Try It

Solving Equations with Fractions

When fractions appear throughout an equation, it is often easiest to clear fractions by multiplying every term by the least common denominator (LCD).

Example 2.3.10

Solve: $\frac{1}{3}x + \frac{1}{6} = 2$

Solution

LCD = 6. Multiply both sides by 6:

$2x + 1 = 12$

Subtract 1:

$2x = 11$

Divide by 2:

$x = \frac{11}{2} \text{ or } 5.5$

✅ Check:

$\frac{1}{3}\left(\frac{11}{2}\right) + \frac{1}{6} = \frac{11}{6} + \frac{1}{6} = 2$

Example 2.3.11

Solve: $\frac{2}{5}x - \frac{1}{10} = \frac{1}{2}x + \frac{3}{10}$

Solution

Multiply both sides by 10:

$4x - 1 = 5x + 3$

Subtract $4x$ from both sides:

$-1 = x + 3$

Subtract 3:

$x = -4$

✅ Check:

Left side: $\frac{2}{5}(-4) - \frac{1}{10} = -\frac{8}{5} - \frac{1}{10} = -\frac{17}{10}$

Right side: $\frac{1}{2}(-4) + \frac{3}{10} = -2 + \frac{3}{10} = -\frac{17}{10}$

Solving Equations with Decimals

You can work with decimals directly or multiply by a power of 10 to clear them.

Example 2.3.12

Solve: $0.6x - 1.4 = 2.8$

Solution

Add 1.4:

$0.6x = 4.2$

Divide by 0.6:

$x = 7$

✅ Check: $0.6(7) - 1.4 = 4.2 - 1.4 = 2.8$

Example 2.3.13

Solve: $0.2x + 0.5 = 0.7x - 1.5$

Solution

Multiply both sides by 10:

$2x + 5 = 7x - 15$

Subtract $2x$ from both sides:

$5 = 5x - 15$

Add 15:

$20 = 5x$

Divide by 5:

$x = 4$

✅ Check:

Left side: $0.2(4) + 0.5 = 0.8 + 0.5 = 1.3$

Right side: $0.7(4) - 1.5 = 2.8 - 1.5 = 1.3$

Try It

Classifying Equations: Conditional, Identity, or Contradiction

Sometimes simplifying eliminates all variables, leaving a true or false statement.

Table 2.3
Type	Description	Example	Result
Conditional	One specific solution	$2x + 3 = 7$	$x = 2$
Identity	True for all $x$	$3x + 6 = 3(x + 2)$	$0 = 0$
Contradiction	False for all $x$	$2(x - 1) = 2x + 3$	$-2 = 3$

Example 2.3.14

Solve: $4x + 8 = 2(2x + 4)$

Solution

Simplify the right side:

$4x + 8 = 4x + 8$

Subtract $4x$ from both sides:

$8 = 8$

✅ Identity (true for all $x$ )

Example 2.3.15

Solve: $5x - 7 = 5x + 3$

Solution

Subtract $5x$ from both sides:

$-7 = 3$

❌ Contradiction (no solution)

Try It

Key Takeaways

Multi-step equations combine addition/subtraction and multiplication/division.
Apply the distributive property before combining like terms.
Clear fractions using the LCD and clear decimals using powers of 10.
If variables disappear:
- True statement → Identity (infinitely many solutions)
- False statement → Contradiction (no solution)
Always check your solution in the original equation.

License

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Launch – Putting It All Together

Solving Two-Step Equations

Variables on Both Sides

Equations Involving Distribution

Solving Equations with Fractions

Solving Equations with Decimals

Classifying Equations: Conditional, Identity, or Contradiction

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