Sometimes systems of equations can be used to model word problems. Let’s jump straight to an example.

**Example:** The school that Matt goes to is selling tickets to a choral performance. On the first day of ticket sales the school sold 12 adult tickets and 3 student tickets for a total of $129. The school took in $104 on the second day by selling 2 adult tickets and 6 student tickets. Find the price of an adult ticket and the price of a student ticket.

**Solution:** Let * a* be the price of an adult ticket, and let

\(12a + 3s = 129\)

Using a similar reasoning, we can model the second day of sales by

\(2a + 6s = 104\)

Combining these two equations gives us a system that we can solve! We use elimination:

\(12a + 3s = 129\)

\(2a + 6s = 104\)

\( - 24a - 6s = - 258\)

\(2a + 6s = 104\)

\( - 22a = - 154\)

\(a = 7\)

That is, an adult ticket cost $7. Then by substituting \(a = 7\) into the second equation, we have

\(2a + 6s = 104\)

\(2\left( 7 \right) + 6s = 104\)

\(14 + 6s = 104\)

\(6s = 90\)

\(s = 15\)

That is, a student ticket costs $15.

**Another Example:** The senior class at High School A and High School B planned separate trips to the water park. The senior class at High School A rented and filled 8 vans and 4 buses with 256 students. High School B rented and filled 4 vans and 6 buses with 312 students. Each van and each bus carried the same number of students. How many students can a van carry? How many students can a bus carry?

**Solution:** Let * v* be the number of students a van can carry. Let

\(8v + 4b = 256\)

Similarly, High School B’s situation can be modeled by

\(4v + 6b = 312\)

We solve the system using elimination

\(8v + 4b = 256\)

\(4v + 6b = 312\)

\(8v + 4b = 256\)

\( - 8v - 12b = - 624\)

\( - 8b = - 368\)

\(b = 46\)

That is, a bus can hold 46 students. Substituting 46 into the first equation gives

\(8v + 4b = 256\)

\(8v + 4\left( {46} \right) = 256\)

\(8v + 184 = 256\)

\(8v = 72\)

\(v = 9\)

That is, each van can hold 9 students.

Below you can **download** some** free** math worksheets and practice.

**Example of one question:**

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**Example of one question:**

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**Example of one question:**

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