1.8Unit 1 Review: Linear Equations
Bring together everything from Unit 1 — one-step through multi-step equations, variables on both sides, fractions, literal equations, and real-world problem solving — in one comprehensive review.
Why This Matters
Equation-solving is the backbone of all future math. A thorough review here ensures you're ready for inequalities, functions, systems, and every unit that follows — and for standardized tests like the SAT.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Unit 1 Big Idea
An equation is a balance. Whatever you do to one side, you must do to the other. Use inverse operations to isolate the variable — then check your answer in the original equation.
Chapter-by-Chapter Summary
One-Step Equations
Apply one inverse operation to isolate the variable.
Two-Step Equations
Undo addition/subtraction first, then multiplication/division.
Multi-Step Equations
Distribute, combine like terms, then solve.
Variables on Both Sides
Collect all variable terms on one side, constants on the other.
Equations with Fractions
Multiply every term by the LCD to clear all fractions, then solve.
Literal Equations & Formulas
Treat all other variables as constants and isolate the target variable.
Problem Solving with Equations
Five steps: Read → Define variable → Write equation → Solve → Check in context.
Unit 1 Key Vocabulary
Equation
A mathematical statement that two expressions are equal.
Solution
The value of the variable that makes the equation true.
Inverse Operations
Operations that undo each other: + and −, × and ÷.
Isolate the Variable
Use inverse operations to get the variable alone on one side.
Like Terms
Terms with the same variable and exponent; can be combined.
Distributive Property
a(b + c) = ab + ac; multiply the factor by each term inside.
Identity
An equation true for all values of the variable (infinite solutions).
No Solution
An equation with no value that makes it true (contradiction).
LCD
Least Common Denominator; used to clear fractions from an equation.
Literal Equation
An equation with two or more variables, solved for one variable.
Consecutive Integers
Integers in order: n, n+1, n+2 (differ by 1).
Consecutive Even/Odd
Even or odd integers in order: n, n+2, n+4 (differ by 2).
Complementary Angles
Two angles whose measures sum to 90°.
Supplementary Angles
Two angles whose measures sum to 180°.
Define the Variable
"Let x = ___" — a clear statement of what the variable represents.
Check in Context
Verify the answer satisfies the original word problem, not just the equation.
Common Mistakes — Unit 1
- Dividing by a negative: Forgetting to flip the sign (relevant in inequalities, but also easy to mis-apply here).
- Distributing incorrectly: 3(x − 4) = 3x − 4 is wrong; it should be 3x − 12.
- Not multiplying every term by the LCD: Missing a constant term when clearing fractions.
- "Less than" reversal: "3 less than x" is x − 3, not 3 − x.
- Consecutive integer gap: Consecutive integers differ by 1; even/odd differ by 2.
- Stopping at x without answering the question: Re-read what the problem actually asks for.
- Skipping the check: Always substitute back into the original equation or problem.
Common Mistakes
Performing the inverse operation on the wrong side — e.g., adding to the variable side instead of both sides.
Whatever you do to one side of the equation, do the exact same thing to the other side.
Distributing to only the first term — e.g., 3(x − 4) = 3x − 4.
Distribute to every term inside the parentheses: 3(x − 4) = 3x − 12.
Not multiplying every term by the LCD when clearing fractions — missing a constant.
Multiply every single term on both sides by the LCD, including constants.
Translating 'three less than x' as 3 − x instead of x − 3.
'Less than' reverses the order: 'three less than x' means x − 3.
Stopping at the value of x without re-reading what the problem asked for.
Always re-read the question. If it asks for the larger number or the total, compute that from x.
Mixed Review — Part A: One-Step & Two-Step Equations
Answers are in the Answer Key section.
x + 14 = 31
y − 9 = −4
−6x = 42
n/5 = −3
2x + 7 = 19
−3y − 4 = 11
x/4 + 2 = 9
5 − 2x = −11
(x + 3)/2 = 8
−x/3 + 1 = 6
Mixed Review — Part B: Multi-Step & Variables on Both Sides
3x + 2x − 7 = 18
4(x − 2) + 3 = 19
2(3x + 1) − 5x = 8
6x + 5 = 3x + 20
4(x + 1) = 2x + 14
5x − 3 = 2(x + 6)
3(2x − 4) = 2(3x − 6)
7x + 2 = 7x − 5
2(x + 5) − 3 = 4x − (2x + 7)
−2(x − 3) + 4x = 3(x − 1) + 5
Mixed Review — Part C: Fractions & Literal Equations
x/3 + 4 = 9
x/2 − x/5 = 6
(2x + 1)/3 = 5
x/4 + x/6 = 5
(3x − 2)/5 = (x + 4)/3
Solve A = (1/2)bh for b.
Solve V = lwh for l.
Solve C = 2πr for r.
Solve ax + by = c for x.
Solve S = (n/2)(a + l) for a.
Mixed Review — Part D: Problem Solving
Four times a number minus 6 is 26. Find the number.
The sum of three consecutive integers is 66. Find the integers.
The sum of two consecutive even integers is 50. Find the integers.
A rectangle has perimeter 52 cm. The length is 4 cm more than twice the width. Find the dimensions.
Two angles are supplementary. One is 3 times the other. Find both angles.
A father is 4 times as old as his son. In 10 years he will be twice as old. Find their current ages.
A taxi charges $3.00 plus $2.50 per mile. A ride cost $18.00. How many miles was the ride?
Almonds cost $5/lb and pecans cost $9/lb. How many pounds of each are needed to make 20 lb of a mix worth $6.60/lb?
Challenge Problems
Solve: (x + 2)/3 − (x − 1)/4 = 2
The sum of three consecutive odd integers is 15 more than twice the largest. Find the integers.
A 15% solution is mixed with a 45% solution to make 120 mL of a 25% solution. How many mL of each?
Two cars start at the same point. Car A travels east at 55 mph; Car B travels west at 65 mph. After how many hours are they 300 miles apart?
Solve for x: a(x + b) = c(x − d). Express x in terms of a, b, c, and d.