4.7Unit Review
A comprehensive review of all Unit 4 topics — graphing, substitution, elimination, special cases, and real-world word problems — followed by a full unit assessment.
Why This Matters
Systems of equations are one of the highest-weighted topics on the SAT. A thorough review here also prepares you for systems of inequalities, matrices in Precalculus, and multi-variable problems in Physics.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Unit 4 Overview
Systems of Equations — Chapters 01 through 06
In this unit you learned to solve systems of two linear equations using three methods, classify systems by their number of solutions (including special cases), and apply systems to real-world word problems. This chapter reviews every key concept and prepares you for the unit assessment.
Unit 4 Concept Map
Unit 4 concept map
Chapter-by-Chapter Summary
Introduction to Systems
- A system = 2 equations, 2 unknowns
- Solution = ordered pair (x, y)
- Verify by substituting into both equations
- Classify: 1 solution / no solution / ∞ solutions
Solving by Graphing
- Convert to y = mx + b
- Plot y-intercept, use slope for second point
- Intersection = solution
- Parallel lines → no solution; same line → ∞ solutions
Solving by Substitution
- Isolate one variable
- Substitute into the other equation
- Solve, then back-substitute
- Best when one variable is already isolated
Solving by Elimination
- Align equations in Ax + By = C form
- Multiply to create opposite coefficients
- Add equations to eliminate one variable
- Best when coefficients match or cancel easily
Applications of Systems
- Define variables clearly (let x = …)
- Write one equation per condition
- Choose the most efficient method
- Interpret answer in context with units
Formula & Strategy Reference Sheet
Keep this reference sheet handy during review
Method Review — Step-by-Step Flowcharts
Solving by graphing — step by step
Solving by substitution — step by step
Solving by elimination — step by step
Choosing the Best Method
Which method should I use?
Is one variable already isolated? (e.g., y = 3x + 1)
Yes → Substitution — plug in directly
Do coefficients of one variable sum to zero or match exactly?
Yes → Elimination — add/subtract directly
Can you easily multiply to create opposite coefficients?
Yes → Elimination — multiply first, then add
Both equations in slope-intercept form?
Yes → Graphing — plot and find intersection
All three methods always work — choose the most efficient one
Method selection decision tree
Three-method strategy comparison
Classifying Systems
System classification reference
Intersecting
1 solution
Parallel
No solution
Coincident
∞ solutions
Three types of systems — graphical interpretation
Graphing Review
1 solution: (2, 3)
No solution (parallel)
Common Mistakes
Solving for x only and forgetting to find y — writing x = 3 as the final answer.
A system solution is an ordered pair (x, y). After finding x, substitute back to find y.
Graphing lines inaccurately and reading the intersection as a non-integer point.
Graphing gives approximate solutions. Use substitution or elimination to confirm exact values.
When using elimination, multiplying only the left side of an equation — e.g., ×2: 6x + 4y = 12 instead of 24.
Multiply every term on both sides by the same factor: 3x + 2y = 12 → 6x + 4y = 24.
Substituting back into the manipulated equation instead of one of the original equations.
Always substitute your answer back into one of the original equations to verify.
Declaring 'no solution' when variables cancel and leave 0 = 0 (infinitely many solutions).
0 = 0 means the equations are the same line — infinitely many solutions. 0 = 5 means no solution.
Common Mistakes Review
Error 1 — Substitution: Forgetting to distribute
✗ This one is actually correct — see Error 2
y = 2x + 3, x + y = 9
x + 2x + 3 = 9
3x + 3 = 9 → x = 2, y = 7 ✓
✓ Distribute before substituting
y = 2(x+1) + 3 = 2x + 5
x + 2x + 5 = 9 → 3x = 4 → x = 4/3
Always distribute fully before combining
Error 2 — Elimination: Forgetting to multiply both sides
✗ Multiplied only the left side
3x + 2y = 12
×2: 6x + 2y = 12 ← WRONG
Only the left side was multiplied
✓ Multiply every term including the right side
3x + 2y = 12
×2: 6x + 4y = 24 ← CORRECT
Multiply every term on both sides
Error 3 — Applications: Not answering the question
✗ Missing context and units
Solved: x = 150, y = 50
Student wrote: "x = 150"
No context, no units, no sentence
✓ Full sentence with labels and units
x = adult tickets = 150
y = student tickets = 50
"There were 150 adult and 50 student tickets sold."
Vocabulary Review
System of Equations
Two or more equations with the same variables, solved simultaneously to find values that satisfy all equations.
Solution of a System
An ordered pair (x, y) that satisfies every equation in the system at the same time.
Consistent System
A system with at least one solution — lines intersect or are the same line.
Inconsistent System
A system with no solution — the lines are parallel and never intersect.
Dependent System
A system with infinitely many solutions — both equations describe the same line.
Substitution Method
Isolate one variable and substitute its expression into the other equation to solve.
Elimination Method
Add or subtract equations (after multiplying if needed) to cancel one variable.
Break-Even Point
The intersection of revenue and cost lines — the point where profit equals zero.
Point of Intersection
The coordinate where two lines cross; the solution when solving by graphing.
Constraint
A condition in a word problem that restricts the possible values of the variables.
Mastery Checklist
Unit 4 Mastery Checklist
Identify a solution as an ordered pair (x, y)
Verify a solution by substituting into both equations
Classify a system as consistent, inconsistent, or dependent
Convert equations to slope-intercept form for graphing
Graph two lines and identify the intersection point
Recognize parallel and coincident lines from graphs
Isolate a variable and substitute into the other equation
Solve the resulting single-variable equation
Multiply equations to create opposite coefficients
Add or subtract equations to eliminate a variable
Define variables and write a system from a word problem
Choose the best method and interpret the answer in context
Check off each skill as you master it
Word Problem Setup Organizer
Word Problem Setup Organizer
Variable 1
Let x = ___________
Variable 2
Let y = ___________
Condition 1 → Equation 1
___ x + ___ y = ___
Condition 2 → Equation 2
___ x + ___ y = ___
Answer sentence:
There were ___ [unit] and ___ [unit].
Use this organizer for every word problem
Quick Review Notes
Graphing
- Convert to y = mx + b first
- Intersection = solution
- Parallel → no solution
- Same line → ∞ solutions
Substitution
- Best when y = … or x = … already
- Substitute the entire expression
- Distribute carefully
- Back-substitute to find the other variable
Elimination
- Align in Ax + By = C form
- Multiply to create opposite coefficients
- Add equations — one variable cancels
- Multiply every term on both sides
Word Problems
- Define variables before writing equations
- Find the two conditions → two equations
- Use a table for coins, tickets, mixtures
- Write a complete answer sentence with units
Mixed Review Problems
Guided Practice Video: Unit 4 Review
Watch the Unit 4 review walkthrough covering systems of equations, then complete the mixed review problems below.
Video by Sang Real Math
Watch on YouTube ↗Solve each system using any method. Show all work and verify your solution.
Solve by graphing: y = x + 1 and y = −2x + 7.
Solve by graphing: y = 3x − 2 and y = 3x + 1. Classify the system.
Solve by substitution: y = 2x − 3 and 3x + y = 12.
Solve by substitution: x = 4y + 1 and 2x − 3y = 7.
Solve by elimination: 2x + y = 8 and x − y = 1.
Solve by elimination: 3x + 2y = 14 and 5x − 2y = 10.
Solve by elimination (multiply first): 2x + 3y = 11 and 4x − y = 7.
Solve by any method: x + y = 10 and 2x − y = 5.
Classify: 4x + 2y = 8 and 2x + y = 4. How many solutions?
Verify whether (3, −1) is a solution to 2x + 3y = 3 and x − y = 4.
Solve: y = −3x + 5 and 6x + 2y = 10. Classify the system.
Solve by substitution: 3x − y = 7 and y = x + 1.
Solve by elimination: 5x + 3y = 19 and 2x − 3y = −5.
Solve by any method: 4x + 5y = 22 and 3x − 2y = 1.
Word problem: Two numbers sum to 48 and their difference is 12. Find both numbers.
Word problem: Tickets cost $6 for adults and $4 for students. 100 tickets were sold for $520. How many of each?
Word problem: Plan A costs $25/month + $0.05/text. Plan B costs $10/month + $0.20/text. At how many texts are the costs equal?
Word problem: A rectangle has perimeter 52 cm. The length is 8 cm more than the width. Find the dimensions.
Word problem: Two cars leave the same point in opposite directions at 55 mph and 45 mph. After how many hours are they 350 miles apart?
Challenge: A plane flies 900 miles with a tailwind in 2 hours and returns against the headwind in 3 hours. Find the plane's speed and the wind speed.
Multiple Choice Review
1. Which ordered pair is a solution to both y = 2x − 1 and y = −x + 5?
(1, 1)
(2, 3)
(3, 5)
(0, 5)
2. A system has no solution. What must be true about the two lines?
They intersect at one point
They are the same line
They are parallel
They are perpendicular
3. Which method is most efficient for: y = 3x + 2 and 2x + y = 12?
Graphing
Substitution
Elimination
All are equally efficient
4. When solving by elimination, you get 0 = 0. What does this mean?
No solution
Exactly one solution
Infinitely many solutions
An error was made
5. Two numbers sum to 30 and one is twice the other. What is the larger number?
10
15
20
25
6. Which system is inconsistent?
y = 2x + 1 and y = 3x − 1
y = 2x + 1 and y = 2x − 3
y = 2x + 1 and 2y = 4x + 2
y = x + 1 and y = −x + 1
7. Solving 3x + y = 10 and x − y = 2 by elimination gives:
x = 2, y = 4
x = 3, y = 1
x = 4, y = −2
x = 1, y = 7
8. A graph shows two lines intersecting at (−2, 5). What is the solution?
x = 5, y = −2
x = −2, y = 5
x = 2, y = −5
x = −5, y = 2
9. Which step comes FIRST when solving by substitution?
Add the two equations
Isolate one variable in one equation
Multiply both equations by the same number
Graph both equations
10. A system is consistent and dependent. How many solutions does it have?
0
1
2
Infinitely many
Short Response Review
Explain in your own words: what does the solution of a system of equations represent graphically?
A student solves a system and gets x = 3, y = 4. How do they verify this is correct?
Write a system of equations that has no solution. Explain how you know it has no solution without solving.
Write a system of equations that has infinitely many solutions. Explain how you know.
When would you choose elimination over substitution? Give a specific example.
A system gives the result 0 = 7 when solved. What does this tell you about the system?
Describe the difference between a consistent and an inconsistent system.
Write a real-world situation that could be modeled by the system: x + y = 50 and 3x + 5y = 190.
Solve 2x + y = 7 and x − y = 2 by elimination. Show every step.
A student says "I can always use graphing to solve any system." Do you agree? Explain.
Challenge Problems
A plane flies 1,200 miles with a tailwind in 3 hours and returns against the headwind in 4 hours. Find the plane's speed in still air and the wind speed.
A chemist mixes a 20% acid solution with a 50% acid solution to make 60 liters of a 30% acid solution. How many liters of each?
Three years ago, a mother was 4 times as old as her son. In 5 years, she will be twice as old. Find their current ages.
A company's revenue is R = 9x and its cost is C = 4x + 250. (a) Find the break-even point. (b) How many units must be sold to earn a profit of at least $200?
Two friends start from towns 240 miles apart and drive toward each other. They meet after 2 hours. One drives 20 mph faster than the other. Find each person's speed.
Real-World Application Problems
Ticket sales: A theater sold 400 tickets for $2,800. Orchestra seats cost $8 and balcony seats cost $5. How many of each type were sold?
Cell plans: Plan A costs $30/month + $0.08/text. Plan B costs $10/month + $0.18/text. For how many texts are the plans equal in cost? Which plan is better for light users?
Coins: A vending machine contains quarters and dimes totaling 35 coins worth $5.75. How many of each coin are there?
Mixture: A nurse mixes a 5% saline solution with a 15% saline solution to make 200 mL of a 9% saline solution. How many mL of each?
Geometry: The perimeter of a rectangle is 64 m. The length is 12 m more than the width. Find the dimensions.
Graphing Practice Grids
Use these grids to graph systems from the review problems above.
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6