# Create Systems of Equations from Word Problems

Some word problems are best solved by creating a** system of equations** (or two equations that use the same variables).

How do you identify those word problems?

**Word problems best solved with a system of equation usually give two different totals.** One total is typically a straight sum (e.g., adult tickets plus kid tickets equal total tickets) and the other is a sum that uses a multiplier (e.g., adult tickets, which cost \$10, plus kid tickets, which cost \$6 equal sum total cost in dollars).

How do you write the equations?

**Start with the straight sum** and assign variables to each item you are adding.

**Then, using those same variables, write the equation with the multipliers.**

Once you have your equations, **solve them like any other system of equations.**

When you're done solving, make sure to pay attention to the question:** which variable does the question want you to find?**

*Example*:

*An extended family of 10 spends \$88 to go to the movies. If adult tickets are \$10 and child tickets are \$4, how many children went to the movies?*

First, figure out what the two totals are:

Total people = 10

Total money = \$88

Next, write an equation for the first total:

adults + children = 10

Decide on your variables: $ a + c = 10$

Now, using those same variables and the multipliers from the question, write the second equation:

$10 \times a + 4 \times c = 88$

$10a + 4c = 88$

Now, you have a system of equations!

$$\require{cancel}\eqalign{&\left. \begin{array}{rcl} &&a+c&&&=10\\ +&&10a+4y&&&=88& \\\hline \end{array}\right\}\quad \text{Write equations, lining up like terms} \\&\left. \begin{array}{rcl} &&-4a+-4c&&&=-40\\ +&&10a+4y&&&=88& \\\hline \end{array}\right\}\quad \text{Multiply top by -4 so you can cancel} \\&\left. \begin{array}{rcl} &&-4a\cancel{-4c}&&&=-40\\ +&&10a\cancel{+4c}&&&=88 \\\hline &&6a&&&=48 \\&&a&&&=8 \end{array}\right\} \quad \text{Add like terms and solve for a}\\&\left. \begin{array}{rcl} &&a+c&&&=10\\ &&8+c&&&=10 \\\hline &&c&&&=2 \end{array}\right\} \quad \text{Plug a into one equation to solve for c}\\&\text{The solution to this system of equations is (8,2)}}$$

**2 children went to the movies. **

Whenever you see a word problem with two totals, consider whether you can write a system of equations. To learn more about solving systems of equation (and more complicated systems of equations) review: Algebra: Systems of Equations (Elimination).

#### Practice Problems:

## Algebra: Create Systems of Equations from Word Problems

For each word problem, set up and solve a system of equations and provide the specific answer requested by the question.

- A family is going on vacation. They will spend 7 nights in Europe, some nights in Paris and some nights in Rome. Their total hotel costs are \$887. If their hotel in Paris costs \$85 per night and their hotel in Rome costs \$158 per night, how many nights are they staying in each location?
- A group of 87 whale watchers spends \$3,584 on tickets for the trip. If adult tickets are \$42 each and senior citizen tickets are \$32 each, how many senior citizens went on the trip?
- You are planting a garden, but you only want peppers and tomatoes (your favorite vegetables). Pepper plants cost 45 cents each. Tomato plants cost 30 cents each. If you buy 20 plants and spend \$7.80, how many of each plant did you buy?
- A student is selling magazines for a school fundraiser and raises \$540. She sold 42 subscriptions. The annual subscriptions went for \$14 each and the quarterly subscriptions went for \$8 each. How many quarterly subscriptions did she sell?
- In a math teacher's grading system, a student earns 10 points for every homework assignment and 40 points for every project. If a student turned in 30 total assignments (homework and projects) and earned 720 points, how many projects did that student do?