Simultaneous Equations
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.
The Rules Every Simultaneous Equation Uses
To solve these problems, you need to turn the story into two separate equations. Follow these four steps to structure your answer every time.
📝 Rule 1: Define Your Unknowns
Assign a variable to each item you need to find. Use letters that help you remember which is which.
Example: "Adults and students" → Let a = adults and s = 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.
3 Worked Examples
One straightforward, one with larger numbers, and one where careful variable choice is key.
1. Museum Tickets
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
Tip: Always check if your answer is a whole number. You can't have 18.5 students!
2. Concert Revenue
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 = VIP tickets
Let s = 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
Tip: Large revenue numbers can be intimidating, but the substitution steps remain exactly the same.
3. The Pizza Choice
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 = small pizzas
Let l = large pizzas
Step 2: Equations.
l + s = 92
25l + 6s = 875
Step 3: Substitute.
Since we want to find s, it's faster to substitute l = 92 - s:
25(92 - s) + 6s = 875
Step 4: Solve.
2300 - 25s + 6s = 875
2300 - 19s = 875
19s = 1425
s = 75
The Trap: If you solve for large pizzas first, you'll get 17. Many students stop there and write "17" as the answer. Always re-read the question to see which item they actually want!
Before You Buy
Want to check the level and layout first? Download the free 3-question sample. It uses the same question style, printable format, and answer-key approach as the full pack.
Download Free Sample PDFGet the Full Practice Pack
The full pack has 90 simultaneous equation word problems across 3 test sets. Perfect for building speed and accuracy with substitution methods. Step-by-step solutions included for every question.
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