From foundations to exam-level word problems. 20 carefully crafted questions — no answer key visible.
20
Questions
4
Methods
★★★
Difficulty range
SUBSTITUTE
Solve one eq. for one var → plug into other
ELIMINATE
Add/subtract eqs. to kill one variable
GRAPH IT
Intersection = solution. Parallel = no solution
CHECK IT
Always plug (x,y) back into BOTH equations
Chapter 1 · Foundations
📖 Worked Example — Substitution
System: y = 2x + 1 and 3x + y = 16
Step 1 · Sub y: 3x + (2x + 1) = 16
Step 2 · Solve: 5x = 15 → x = 3
Step 3 · Back-sub: y = 2(3)+1 = 7
✓ Solution: (3, 7)
01
substitution
Solve the system using substitution.
{
y = x + 4
2x + y = 10
Substitute y = x + 4 directly into 2x + y = 10. You'll get 2x + (x+4) = 10.
02
elimination
Solve by elimination (add the equations).
{
x + y = 9
x − y = 3
Add both equations: the y terms cancel. 2x = 12 → x = 6. Then find y.
03
graphingconcept
Two lines are graphed. Line 1: y = 2x − 1. Line 2: y = −x + 5. At what point do they intersect?
Set both y-expressions equal: 2x−1 = −x+5. Solve for x, then substitute back.
Chapter 2 · Word Problems
📖 Worked Example — Word Problem Setup
Problem: "Two numbers add to 20. Their difference is 4."
Define: x = larger, y = smaller
Translate: x + y = 20 and x − y = 4
Solve (add): 2x = 24 → x = 12, y = 8
04
word problemmedium
Real World
A coffee shop sells small cups for $3 and large cups for $5. One morning, they sold 60 cups total and made $220. How many large cups were sold?
Let s = small, L = large. Write: s + L = 60 and 3s + 5L = 220. Solve by elimination.
05
word problemmedium
Real World
The sum of two numbers is 48. One number is three times the other. What are the two numbers?
Let x and y be the numbers. x + y = 48 and x = 3y. Substitute.
06
word problemmedium
Real World
A gym charges a $20 sign-up fee plus $15/month. A rival gym charges no sign-up fee but $25/month. After how many months will the total cost be equal?
Gym A: C = 20 + 15m. Gym B: C = 25m. Set equal and solve for m.
Chapter 3 · Special Cases
📖 Key Concept — No Solution vs. Infinite Solutions
No solution (parallel lines): same slope, different intercept
e.g. y = 2x + 1 and y = 2x + 5 → never intersect
Infinite solutions (same line): identical equations after simplification
e.g. y = 2x + 1 and 2y = 4x + 2 → same line
07
special casesmedium
How many solutions does this system have?
{
2x + 4y = 8
x + 2y = 4
08
special casesmedium
Which statement best describes this system?
{
y = 3x − 2
y = 3x + 5
Chapter 4 · Elimination with Multiplication
📖 Worked Example — Multiply Before Eliminating
System: 2x + 3y = 12 and 5x + 2y = 11
Multiply eq.1 by 2: 4x + 6y = 24
Multiply eq.2 by 3: 15x + 6y = 33
Subtract: 11x = 9 → x = 9/11 (messy → keep going)
Better: multiply eq.1 by 5, eq.2 by 2 → eliminate x instead
09
elimination ×hard
Solve by multiplying to create matching coefficients.
{
3x + 2y = 16
2x − y = 6
Multiply eq.2 by 2 to get −2y. Add to eq.1 to eliminate y. Then back-substitute.
10
elimination ×hard
Solve the system.
{
4x + 3y = 10
2x − 5y = −8
Multiply eq.2 by 2: 4x − 10y = −16. Subtract from eq.1 to eliminate x. Solve for y first.
Chapter 5 · Exam-Level Word Problems
11
word problemhard
Real World
A boat travels 60 miles downstream in 3 hours and the same 60 miles upstream in 5 hours. Let b = boat speed in still water and c = current speed. What is the speed of the current?
A chemist mixes a 20% acid solution with a 50% acid solution to get 300 mL of a 30% solution. How many mL of the 50% solution is needed?
Let x = mL of 20%, y = mL of 50%. Equations: x+y=300 and 0.20x + 0.50y = 0.30×300 = 90.
13
word problemhard
Real World
Two trains leave different cities 360 miles apart, heading toward each other. Train A travels at 80 mph, Train B at 100 mph. After how many hours do they meet?
Distance A + Distance B = 360. So 80t + 100t = 360. Solve for t.
14
word problemhard
Real World
The perimeter of a rectangle is 56 cm. The length is 4 more than twice the width. Find the length.
Perimeter: 2(l + w) = 56 → l + w = 28. Second equation: l = 2w + 4. Substitute.
Chapter 6 · Trap Questions — Most Missed
15
trap ⚠️fractions
Solve the system. Watch out for fractions!
{
x/2 + y/3 = 4
x/4 − y/6 = 1
Clear fractions! Multiply eq.1 by 6: 3x + 2y = 24. Multiply eq.2 by 12: 3x − 2y = 12. Then add.
16
trap ⚠️negative coeff
Solve carefully — negative signs are the trap here.
{
−3x + 2y = −1
6x − 4y = 2
Multiply eq.1 by 2: −6x + 4y = −2. Add to eq.2: 0 = 0. What does that tell you?
17
trap ⚠️word problem
Real World
At a school fair, adult tickets cost $6 and child tickets cost $3. A total of 200 tickets were sold for $900. A student claims "100 of each were sold." Is the student correct?
Check: 100×$6 + 100×$3 = $600 + $300 = $900. The total IS $900. Don't overthink it!
18
trap ⚠️rearrange first
The equations look messy — rearrange to standard form first!
{
3y − x = 2x + 5
2x + y − 3 = 0
Eq.1 → −3x + 3y = 5. Eq.2 → 2x + y = 3. Now solve by substitution or elimination.
19
exam-levelword problem
Real World
A store sells notebooks for $2 each and pens for $1.50 each. Maya buys 12 items and spends $20.50. Her friend thinks she bought 7 notebooks. How many notebooks did Maya actually buy?
n + p = 12 and 2n + 1.5p = 20.5. Multiply eq.1 by 1.5: 1.5n + 1.5p = 18. Subtract to find n.
20
exam boss 🔥age problem
Real World
Five years ago, a father was 4 times as old as his son. In 3 years, he will be only 2 times as old. What is the father's current age?
Let f = father now, s = son now. Five yrs ago: (f−5) = 4(s−5). In 3 yrs: (f+3) = 2(s+3). Solve!