Simultaneous Equations Practice Test Worksheets
Simultaneous equations often appear as word problems involving two different types of items and a total cost. By learning how to translate these scenarios into algebra, you can solve for both unknowns with perfect accuracy.
For students who can solve simple equations, but get stuck when two unknowns have to be linked and eliminated.
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What Do Simultaneous Equations Questions Look Like?
Simultaneous equations word problems present scenarios with two unknown quantities linked by two separate conditions, such as total count and total revenue. Solving them requires setting up both equations and eliminating one variable. Here is a worked example.
Build and Solve Two Equations
Question: A museum adult ticket costs $8 while a student ticket costs $5. Forty-seven visitors entered the museum and the total admissions collected were $322. How many students visited the museum?
Worked Method
Step 1: Define variables.
Let s = number of students
Let a = number of adults
Step 2: Create equations.
Total visitors: a + s = 47
Total revenue: 8a + 5s = 322
Step 3: Substitute.
From the first equation, a = 47 - s.
Substitute this into the second: 8(47 - s) + 5s = 322
Step 4: Solve.
376 - 8s + 5s = 322
376 - 3s = 322
3s = 54
s = 18
Step 5: Check both equations.
There are 47 − 18 = 29 adults.
Count: 29 + 18 = 47.
Revenue: 8 × 29 + 5 × 18 = 232 + 90 = 322.
Answer: 18 students visited the museum.
Tip: Counts must be non-negative whole numbers, but that alone does not prove the answer. Check both the total count and the total revenue.
Rules to Help You Solve Simultaneous Equations Questions
To solve simultaneous equations like the example above, you need to turn the story into two distinct algebraic relationships. Follow these four rules:
📝 Rule 1: Define Your Unknowns
Assign a variable to the number of each type of item or person. Keep these counts distinct from the price per item or person.
Example: "Adults and students" → Let a = number of adults and s = number of students.
🔢 Rule 2: The "Total Count"
Look for the total number of items sold or people who attended. This gives you your first, simplest equation.
Example: "47 visitors in total" → a + s = 47.
💰 Rule 3: The "Total Value"
Multiply each variable by its cost or value to match the total money collected.
Example: "$8 per adult, $5 per student, total $322" → 8a + 5s = 322.
⚖️ Rule 4: Substitution
Rearrange Equation 1 and substitute it into Equation 2. This leaves you with just one variable to solve.
Example: a = 47 - s → 8(47 - s) + 5s = 322.
Two More Worked Examples
Here are two more examples showing how to manage larger revenue totals and avoid solving for the wrong unknown.
Substitute to Find the Required Count
Question: A VIP concert ticket costs $20 while a standard ticket costs $3. One hundred and two fans attended the concert and the total ticket revenue was $1428. How many VIP tickets were sold?
Worked Method
Step 1: Define variables.
Let v = number of VIP tickets
Let s = number of standard tickets
Step 2: Create equations.
v + s = 102
20v + 3s = 1428
Step 3: Substitute.
Substitute s = 102 - v into the second equation:
20v + 3(102 - v) = 1428
Step 4: Solve.
20v + 306 - 3v = 1428
17v = 1122
v = 66
Step 5: Check both equations.
There are 102 − 66 = 36 standard tickets.
Count: 66 + 36 = 102.
Revenue: 20 × 66 + 3 × 36 = 1320 + 108 = 1428.
Answer: 66 VIP tickets were sold.
Tip: Large revenue numbers can be intimidating, but the substitution steps remain exactly the same.
Answer for the Correct Variable
Question: A large pizza costs $25 while a small pizza costs $6. A restaurant sold ninety-two pizzas and the total sales were $875. How many small pizzas were sold?
Worked Method
Step 1: Define variables.
Let s = number of small pizzas
Let l = number of large pizzas
Step 2: Equations.
l + s = 92
25l + 6s = 875
Step 3: Substitute.
Substitute l = 92 - s to leave only the requested variable, s, in the equation:
25(92 - s) + 6s = 875
Step 4: Solve.
2300 - 25s + 6s = 875
2300 - 19s = 875
19s = 1425
s = 75
Step 5: Check both equations.
There are 92 − 75 = 17 large pizzas.
Count: 17 + 75 = 92.
Revenue: 25 × 17 + 6 × 75 = 425 + 450 = 875.
Answer: 75 small pizzas were sold.
The Trap: If you solve for large pizzas first, you'll get 17. That is not the requested count: subtract it from 92 to find 75 small pizzas. Always check which variable the question asks for.
