### Word problems

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 s represent the price of a student ticket.  On the first day of the performance the 12 adult tickets were sold at the price of a and 3 student tickets were sold at the price of s.  The sum of their sales was $129. We can model this by $$12a + 3s = 129$$ 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 b be the number of students a bus can carry.  High School A’s situation can be modeled by

$$8v + 4b = 256$$

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.

16364 x

This free worksheet contains 10 assignments each with 24 questions with answers.

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11104 x

This free worksheet contains 10 assignments each with 24 questions with answers.

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9252 x

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

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### Geometry

Circles
Congruent Triangles
Constructions
Parallel Lines and the Coordinate Plane
Properties of Triangles

### Algebra and Pre-Algebra

Beginning Algebra
Beginning Trigonometry
Equations
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Linear Equations and Inequalities
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