# 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?

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:

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).

• ## 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.

1. 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? 2. 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?
3. 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? 4. 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?
5. 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?