4.6Systems of Word Problems
Translate real-world situations into systems of equations and solve them using the most appropriate method — graphing, substitution, or elimination.
Why This Matters
Real-world problems rarely have just one unknown. Mixture problems, break-even analysis, and motion problems all require systems — skills used daily in business, engineering, and science careers.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How can a system of two equations model a real-world situation, and how do we choose the most efficient method to find the solution?
Lesson Overview
Systems of equations are powerful tools for solving real-world problems that involve two unknown quantities and two conditions. The key skill is translation — reading a word problem carefully, identifying the two unknowns, and writing one equation for each condition. Once the system is set up, we choose the most efficient method: graphing for a visual overview, substitution when one variable is already isolated, or elimination when coefficients are opposites or equal. The final step is always to interpret the answer in context — including units and a complete sentence.
6-step word-problem solving process
Translating words into equations
Ticket sales: 200 adult, 100 student tickets
Define Variables First
Let a = number of adult tickets
Let s = number of student tickets
a + s = 300
8a + 5s = 2100
Which method should I use?
Is one variable already isolated?
Yes → Substitution
Do coefficients of one variable sum to zero or match?
Yes → Elimination
Both equations in slope-intercept form?
Yes → Graphing
Multiply to create opposite coefficients → Elimination
Decision tree: choosing the best method
Error Analysis
✗ Wrong — forgot to answer the question
x + y = 20, 3x + 5y = 80
Solved: x = 10, y = 10
Student wrote: "x = 10" — but what does x represent?
✓ Correct — answer the question with context
x = adult tickets = 10
y = student tickets = 10
10 adult tickets and 10 student tickets were sold.
Always answer in context — include units and labels
Worked Examples
Ticket sales: A school sold 300 tickets for a total of $2,100. Adult tickets cost $8 and student tickets cost $5. How many of each type were sold?
Let a = adult tickets, s = student tickets.
Condition 1 (count): a + s = 300.
Condition 2 (value): 8a + 5s = 2100.
Multiply Eq 1 by 5: 5a + 5s = 1500.
Subtract from Eq 2: 3a = 600 → a = 200.
Back-sub: 200 + s = 300 → s = 100.
Verify: 200+100=300 ✓; 8(200)+5(100)=2100 ✓
Quantity × Value = Total organizer
Graph confirms 200 adult, 100 student tickets
Cell phone plans: Plan A costs $30/month plus $0.05 per minute. Plan B costs $10/month plus $0.15 per minute. For how many minutes do the plans cost the same? Which plan is cheaper for heavy users?
Let m = minutes used, C = monthly cost.
Plan A: C = 30 + 0.05m. Plan B: C = 10 + 0.15m.
Set equal: 30 + 0.05m = 10 + 0.15m.
20 = 0.10m → m = 200 minutes.
Cost at 200 min: C = 30 + 0.05(200) = $40.
For more than 200 minutes, Plan A is cheaper (lower per-minute rate).
Break-even at 200 minutes
Distance–Rate–Time: Two cars leave the same point traveling in opposite directions. Car A travels at 55 mph and Car B at 45 mph. After how many hours are they 300 miles apart?
Let t = time in hours (same for both cars).
Distance A: d_A = 55t. Distance B: d_B = 45t.
Total distance: d_A + d_B = 300.
55t + 45t = 300 → 100t = 300 → t = 3 hours.
Verify: 55(3) + 45(3) = 165 + 135 = 300 ✓
d = r × t organizer
Verify t = 3 hrs
Equation 1
55(3)=165
165+135=300 ✓
Equation 2
45(3)=135
Total=300 ✓
Both satisfied ✓
Coins: A piggy bank contains quarters and dimes. There are 24 coins worth $3.75 total. How many of each coin are there?
Let q = quarters, d = dimes.
Condition 1 (count): q + d = 24.
Condition 2 (value): 0.25q + 0.10d = 3.75 → multiply by 100: 25q + 10d = 375.
Multiply Eq 1 by 10: 10q + 10d = 240.
Subtract: 15q = 135 → q = 9.
Back-sub: 9 + d = 24 → d = 15.
Verify: 9+15=24 ✓; 0.25(9)+0.10(15)=2.25+1.50=3.75 ✓
Quantity × Value = Total organizer
Mixture: A chemist needs 40 liters of a 35% acid solution. She has a 20% solution and a 50% solution. How many liters of each should she mix?
Let x = liters of 20% solution, y = liters of 50% solution.
Condition 1 (volume): x + y = 40.
Condition 2 (acid): 0.20x + 0.50y = 0.35(40) = 14.
Multiply Eq 1 by 0.20: 0.20x + 0.20y = 8.
Subtract: 0.30y = 6 → y = 20.
Back-sub: x = 40 − 20 = 20.
Verify: 20+20=40 ✓; 0.20(20)+0.50(20)=4+10=14 ✓
Acid Mixture
20%
x liters
50%
y liters
35%
40 liters
Volume equation + Concentration equation
Guided Practice
Guided Practice Video: Systems of Equations Word Problems
Watch the guided practice walkthrough for systems of equations word problems, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Ticket sales: A theater sold 250 tickets for $1,700. Orchestra seats cost $8 and balcony seats cost $5. How many of each type were sold?
Hint: Let o = orchestra, b = balcony. Write two equations: one for total tickets, one for total revenue. Use elimination — multiply the count equation by 5.
Cell plans: Plan X costs $25/month plus $0.10 per text. Plan Y costs $15/month plus $0.20 per text. For how many texts do the plans cost the same?
Hint: Write C = 25 + 0.10t for Plan X and C = 15 + 0.20t for Plan Y. Set them equal and solve for t.
Distance–Rate–Time: A boat travels 60 miles downstream in 3 hours and 60 miles upstream in 5 hours. Find the speed of the boat in still water and the speed of the current.
Hint: Let b = boat speed, c = current speed. Downstream: b + c = 20. Upstream: b − c = 12. Add the equations.
Mixture: A farmer mixes a 10% fertilizer solution with a 40% fertilizer solution to make 30 gallons of a 20% solution. How many gallons of each does he use?
Hint: Let x = gallons of 10%, y = gallons of 40%. Write: x + y = 30 and 0.10x + 0.40y = 0.20(30) = 6.
Geometry: Two angles are supplementary (sum to 180°). One angle is 30° more than twice the other. Find both angles.
Hint: Let x and y be the angles. Write: x + y = 180 and x = 2y + 30. Substitute the second equation into the first.
Key Vocabulary
System of Equations
Two or more equations with the same variables, solved simultaneously to find values that satisfy all equations.
Define the Variables
Clearly stating what each variable represents before writing equations — e.g., 'Let x = number of adult tickets.'
Two Conditions
The two separate facts or constraints in a word problem that each produce one equation.
Break-Even Point
The point where revenue equals cost — the intersection of the revenue and cost lines on a graph.
Mixture Problem
A problem involving combining two substances of different concentrations or values to produce a mixture with a known total.
Distance–Rate–Time
The relationship d = r × t, used to set up equations for problems involving travel at different speeds.
Interpret the Solution
Translating the algebraic answer (x, y) back into the context of the problem — answering the original question with units.
Constraint
A condition or limitation in a problem that restricts the possible values of the variables.
Interactive Practice — 5 Questions
A school sold adult tickets for $6 and student tickets for $3. They sold 200 tickets for $900. Which system correctly models this?
Two numbers sum to 40 and their difference is 12. What are the two numbers?
Plan A: $15/month + $0.10/text. Plan B: $5/month + $0.30/text. At how many texts do the plans cost the same?
A graph shows two cost lines intersecting at (50, 30). What does this intersection represent?
Two cars leave the same point in opposite directions at 55 mph and 45 mph. After how many hours are they 400 miles apart?
Independent Practice
Independent Practice
Ticket sales: 400 tickets were sold for $2,600. Adult tickets cost $8 and child tickets cost $3. How many of each?
Shopping: Store A charges $10 per book plus a $4 fee. Store B charges $12 per book with no fee. For how many books is the cost the same?
Two numbers sum to 52 and their difference is 14. Find both numbers.
Coins: A jar has nickels and quarters totaling 30 coins worth $4.50. How many of each?
Cell plans: Plan A is $20/month + $0.08/min. Plan B is $8/month + $0.20/min. At how many minutes are the costs equal?
Common Mistakes
Setting up one equation for a two-variable problem — using only one unknown when two are needed.
Identify both unknowns, define them with variables, and write a separate equation for each constraint.
Mixing up which variable represents which quantity — e.g., letting x = cost of item B when you defined x = cost of item A.
Write your variable definitions clearly at the start: 'Let x = … and y = …' and refer back to them.
Solving for x and y but not answering the actual question — e.g., the problem asks for the total and you only report x.
Re-read the question after solving. Compute the requested quantity from your x and y values.
Forgetting to include units in the final answer for a real-world problem.
Always attach units to your answer: dollars, hours, pounds, etc. A number without units is incomplete.
Math Tips
Always define your variables first — write "Let x = …" and "Let y = …" before writing any equation.
Every application problem has exactly two conditions — find both and write one equation for each.
Use a table or organizer for coin, ticket, mixture, and d = rt problems to prevent setup errors.
After solving, re-read the problem and verify your answer makes sense in context (no negative quantities).
Write a complete answer sentence with units — "There were 200 adult tickets and 100 student tickets sold."