4.2 Solving One-Step & Multi-Step Linear Equations

4.2 Solving One-Step & Multi-Step Linear Equations

An equation is a mathematical statement that two expressions are equal. Solving an equation means finding all values of the variable that make the equation true—these values are called solutions or roots of the equation. On the Praxis 5003 exam, candidates are expected to demonstrate fluency with solving linear equations in one variable, including equations with fractions, decimals, parentheses, and variables on both sides.

The Concept of Equality & Inverse Operations

The fundamental model for understanding linear equations is the balanced scale. Whatever operation is performed on one side of an equation must be performed on the other side to preserve equality. This foundational concept relies on the Properties of Equality:

• Addition Property of Equality: If $a = b$, then $a + c = b + c$.
• Subtraction Property of Equality: If $a = b$, then $a - c = b - c$.
• Multiplication Property of Equality: If $a = b$, then $a \cdot c = b \cdot c$.
• Division Property of Equality: If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.

To isolate a variable, we apply inverse operations (operations that undo each other). Addition undos subtraction, and multiplication undos division. When solving equations, inverse operations are applied in the reverse order of operations (often remembered as SADMEP: Subtraction/Addition, Division/Multiplication, Exponents, Parentheses).

One-Step and Two-Step Linear Equations

• One-Step Equations: Solved using a single inverse operation.
  • Example: $x + 7 = 15 \implies x = 15 - 7 = 8$.
  • Example: $-4x = 28 \implies x = \frac{28}{-4} = -7$.
• Two-Step Equations: Require two inverse operations—typically removing the constant term first, followed by eliminating the variable coefficient.

Step-by-Step Worked Example 1: Two-Step Equation

Problem: Solve $-5x + 14 = -31$ for $x$.

Solution:

1. Isolate the variable term by subtracting $14$ from both sides: $-5x + 14 - 14 = -31 - 14$ $-5x = -45$

2. Isolate $x$ by dividing both sides by $-5$: $\frac{-5x}{-5} = \frac{-45}{-5}$ $x = 9$

3. Check the solution: $-5(9) + 14 = -45 + 14 = -31 \quad \checkmark$

Final Answer: $x = 9$

Multi-Step Equations: Clearing Fractions & Parentheses

When linear equations contain parentheses or rational expressions (fractions), applying systematic techniques simplifies computation and prevents errors.

Strategy for Clearing Fractions

To eliminate fractions entirely, find the Least Common Denominator (LCD) of all denominators present in the equation, then multiply every single term on both sides of the equation by that LCD.

Step-by-Step Worked Example 2: Clearing Fractions & Solving

Problem: Solve for $x$: $\frac{3}{4}x - \frac{1}{2} = \frac{1}{3}x + 2$.

Solution:

1. Identify the LCD of denominators $4$, $2$, and $3$:

   • Denominators: $4, 2, 3 \implies \text{LCD} = 12$.

2. Multiply every term on both sides by $12$: $12 \cdot \left(\frac{3}{4}x\right) - 12 \cdot \left(\frac{1}{2}\right) = 12 \cdot \left(\frac{1}{3}x\right) + 12 \cdot (2)$

3. Simplify each term:

   • $12 \cdot \frac{3}{4}x = 9x$
   • $12 \cdot \frac{1}{2} = 6$
   • $12 \cdot \frac{1}{3}x = 4x$
   • $12 \cdot 2 = 24$

   The equation becomes: $9x - 6 = 4x + 24$

4. Collect variable terms on the left side (subtract $4x$): $9x - 4x - 6 = 24 \implies 5x - 6 = 24$

5. Collect constant terms on the right side (add $6$): $5x = 30$

6. Divide by $5$: $x = 6$

7. Check in original equation: $\frac{3}{4}(6) - \frac{1}{2} = \frac{18}{4} - \frac{2}{4} = \frac{16}{4} = 4$ $\frac{1}{3}(6) + 2 = 2 + 2 = 4 \quad \checkmark$

Final Answer: $x = 6$

Solution Classifications: Conditional, Identity, Contradiction

Not all linear equations have exactly one solution. Linear equations fall into one of three classifications based on their solution sets:

Examples of Special Cases

• Identity Example: Solve $3(2x + 4) = 6x + 12$.

  • Expand left side: $6x + 12 = 6x + 12$.
  • Subtract $6x$: $12 = 12$.
  • Since $12 = 12$ is always true, the equation is an identity with infinitely many solutions.

• Contradiction Example: Solve $4(x - 2) = 4x + 9$.

  • Expand left side: $4x - 8 = 4x + 9$.
  • Subtract $4x$: $-8 = 9$.
  • Since $-8 = 9$ is false, the equation is a contradiction with no solution.

Word Problem Setups: Translating Contexts to Equations

Solving real-world mathematical problems requires creating an equation model from text descriptions.

Step-by-Step Worked Example 3: Contextual Word Problem

Problem: A school fundraiser sold a total of $450$ tickets for a theater production. Student tickets cost $$4$ each and adult tickets cost $$7$ each. If the total revenue raised was $$2,370$, how many adult tickets were sold?

Solution:

1. Define the variables:

   • Let $a$ = number of adult tickets sold.
   • Total tickets = $450$, so student tickets = $450 - a$.

2. Set up the revenue equation: $\text{Student Revenue} + \text{Adult Revenue} = \text{Total Revenue}$ $4(450 - a) + 7a = 2,370$

3. Distribute and combine like terms: $1,800 - 4a + 7a = 2,370$ $1,800 + 3a = 2,370$

4. Solve for $a$: $3a = 2,370 - 1,800$ $3a = 570 \implies a = 190$

5. Interpret the result and check:

   • Adult tickets ($a$) = $190$
   • Student tickets ($450 - 190$) = $260$
   • Check revenue: $4(260) + 7(190) = 1,040 + 1,330 = 2,370 \quad \checkmark$

Final Answer: $190$ adult tickets

Praxis Exam Traps & Common Candidate Errors

1. Distributing to only the first term: In $3(x + 4) = 19$, writing $3x + 4 = 19$ instead of $3x + 12 = 19$.
2. Sign errors during subtraction across the equals sign: Subtracting a negative number becomes addition. Double check signs when moving terms across equal signs.
3. Forgetting to multiply all terms when clearing fractions: When clearing fractions by multiplying by the LCD, candidates frequently forget to multiply constant terms that do not have denominators.