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).