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Unit 1: Linear Relationships and Equations

Topic 3

Multi-Step and Fractional Equations

Many SAT algebra questions involve equations that take more than one step to solve. These are called multi-step equations. They may include parentheses, fractions, or variables on both sides.
The goal is still the same: isolate the variable. To do that, we simplify both sides of the equation by:
  1. Using the distributive property to remove parentheses.
  2. Combining like terms on each side.
  3. Using addition, subtraction, multiplication, or division to isolate the variable.
When fractions appear, it is often easiest to eliminate them first by multiplying both sides by a common denominator.

Core Skills

  • Use the distributive property to expand expressions.
  • Combine like terms carefully.
  • Multiply through by the least common denominator (LCD) to remove fractions.
  • Check your final answer by substituting it back into the equation.

Example 1: Multi-Step Equation

Solve for xx:
3(x−2)=93(x - 2) = 9
Step 1: Distribute.
3x−6=93x - 6 = 9
Step 2: Add 6 to both sides.
3x=153x = 15
Step 3: Divide by 3.
x=5x = 5
Check:
3(5−2)=9⇒3(3)=9✓3(5 - 2) = 9 \Rightarrow 3(3) = 9 \quad \checkmark
x=5\boxed{x = 5}

Example 2: Fractional Equation

Solve for xx:
2x−35=3\dfrac{2x - 3}{5} = 3
Step 1: Multiply both sides by 5 to remove the denominator.
2x−3=152x - 3 = 15
Step 2: Add 3 to both sides.
2x=182x = 18
Step 3: Divide by 2.
x=9x = 9
Check:
2(9)−35=18−35=155=3✓\dfrac{2(9) - 3}{5} = \dfrac{18 - 3}{5} = \dfrac{15}{5} = 3 \quad \checkmark
x=9\boxed{x = 9}

Key Takeaways

  • Simplify both sides before isolating the variable.
  • Use the distributive property correctly when parentheses are involved.
  • When fractions are present, multiply by the LCD to clear denominators.
  • Always check your answer to avoid sign or arithmetic errors.