Concept Overview
A multi-step equation requires more than one operation to isolate the variable. These equations often involve distributing, combining like terms, and performing multiple inverse operations.
The process still relies on the principle of equality — whatever is done to one side must be done to the other — but now in a series of logical steps.
Key Vocabulary
- Equation: A mathematical statement showing that two expressions are equal, using an equals sign ( = )
- Distribute: To multiply a term outside parentheses across each term inside (e.g., \( a(b + c) = ab + ac \))
- Combine Like Terms: To simplify an expression by adding or subtracting terms with the same variable and exponent
- Inverse Operation: An operation that reverses the effect of another (e.g., addition ↔ subtraction, multiplication ↔ division)
- Isolate the Variable: To get the variable by itself on one side of the equation
- Simplify: To reduce an expression or equation to its most compact and clear form
- Balance the Equation: To keep both sides equal by performing the same operation on both sides
Steps for Solving Multi-Step Equations
- Distribute: If the equation contains parentheses, apply the distributive property to remove them. This creates a clearer expression to work with.
- Combine Like Terms: Simplify each side of the equation by combining terms with the same variable or constant value.
- Move Terms: Use inverse operations to get all variable terms on one side and constant terms on the other. Choose a side that avoids negative coefficients if possible.
- Isolate the Variable: Undo the operation next to the variable using the opposite (inverse) operation — such as dividing if it’s being multiplied.
- Simplify and Solve: Finish solving and write your solution. You should now have a single value for the variable.
- Check Your Solution: Substitute your result back into the original equation to verify it works. Both sides should be equal.
While these steps outline a reliable process, it’s important to note that many equations can be solved in more than one way. Some methods may be more efficient or help avoid negative values or fractions. With practice, these strategies become more intuitive.
Worked Examples
Example 1:
Problem: \( 2x + 3 + x = 12 \)
Step 1: Combine like terms on the left side:
\( (2x + x) + 3 = 12 \Rightarrow 3x + 3 = 12 \)
Step 2: Subtract 3 from both sides:
\( 3x = 9 \)
Step 3: Divide both sides by 3:
\( x = 3 \)
Check: Original equation: \( 2x + 3 + x = 12 \)
Substitute \( x = 3 \): \( 2(3) + 3 + 3 = 6 + 3 + 3 = 12 \) ✅
Example 2:
Problem: \( 5(x – 1) + 3 = 23 \)
Step 1: Distribute \( 5 \) to both terms inside parentheses:
\( 5x – 5 + 3 = 23 \)
Step 2: Combine like terms:
\( 5x – 2 = 23 \)
Step 3: Add 2 to both sides:
\( 5x = 25 \)
Step 4: Divide both sides by 5:
\( x = 5 \)
Check: \( 5(5 – 1) + 3 = 5(4) + 3 = 20 + 3 = 23 \) ✅
Example 3:
Problem: \( 6x + 4 = 3x + 19 \)
Step 1: Subtract \( 3x \) from both sides:
\( 3x + 4 = 19 \)
Step 2: Subtract 4 from both sides:
\( 3x = 15 \)
Step 3: Divide both sides by 3:
\( x = 5 \)
Check: \( 6(5) + 4 = 30 + 4 = 34 \); \( 3(5) + 19 = 15 + 19 = 34 \) ✅
Example 4:
Problem: \( 2x + 7 = x – 5 \)
Step 1: Subtract \( x \) from both sides:
\( x + 7 = -5 \)
Step 2: Subtract 7 from both sides:
\( x = -12 \)
Check: LHS: \( 2(-12) + 7 = -24 + 7 = -17 \), RHS: \( -12 – 5 = -17 \) ✅
Example 5:
Problem: \( \frac{1}{2}(x + 6) = \frac{3}{4}x \)
Step 1: Eliminate fractions by multiplying both sides by 4:
\( 4 \cdot \frac{1}{2}(x + 6) = 4 \cdot \frac{3}{4}x \Rightarrow 2(x + 6) = 3x \)
Step 2: Distribute on the left side:
\( 2x + 12 = 3x \)
Step 3: Subtract \( 2x \) from both sides:
\( 12 = x \)
Check: LHS: \( \frac{1}{2}(12 + 6) = \frac{1}{2}(18) = 9 \), RHS: \( \frac{3}{4}(12) = 9 \) ✅
Common Mistakes
- Forgetting to distribute a negative sign
- Combining unlike terms (e.g., \( 3x + 4 \) as \( 7x \))
- Dropping a term when moving between steps
Try It Yourself!
- Solve: \( 2(x + 3) = 14 \)
- Solve: \( 4x – 7 = 3x + 5 \)
- Solve: \( 5x + 2 = 3(x + 4) \)
- Solve: \( \frac{1}{2}x – 3 = \frac{1}{4}x + 1 \)
Reveal Answers
- Distribute: \( 2x + 6 = 14 \) → Subtract 6 → \( 2x = 8 \) → \( x = 4 \)
- Subtract \( 3x \): \( x – 7 = 5 \) → Add 7 → \( x = 12 \)
- Expand right side: \( 5x + 2 = 3x + 12 \) → Subtract \( 3x \): \( 2x + 2 = 12 \) → Subtract 2 → \( 2x = 10 \) → \( x = 5 \)
- Multiply all terms by 4: \( 2x – 12 = x + 4 \) → Subtract \( x \): \( x – 12 = 4 \) → Add 12 → \( x = 16 \)
Connections & Extensions
Multi-step equations appear in geometry, physics, and budgeting problems. Understanding this process prepares for solving inequalities, systems of equations, and function-based modeling.