Systems of Equations Word Problems
Model real-world situations with two unknowns using systems of equations. Cover mixture problems, rate problems, cost/revenue problems, and number problems. Practice choosing the most efficient solution method.
A lesson of Algebra 1, in Systems of Equations & Inequalities, about 30 minutes.
Systems in Words
By the end of this lesson, you will be able to spot when a problem needs two equations, define both variables precisely, build the system, solve it by whichever method fits, and answer the question that was asked.
Two Unknowns Need Two Facts
One unknown needs one equation. Two unknowns need two.
The signal is a problem with two quantities you cannot work out separately, and two separate statements tying them together.
Adult tickets are $$8$ and child tickets are $$5$. Forty tickets were sold for $$260$.
Two unknowns: how many adults, how many children. Two facts: the count, and the money. That is a system.
$$\begin{cases} a + c = 40 \ 8a + 5c = 260 \end{cases}$$
The two equations usually count different things, and noticing which is which is the whole setup. One counts TICKETS, the other counts DOLLARS. Writing both as if they counted the same thing is the error that produces a wrong system that solves perfectly.
The Setup, in Order
- Name both variables, with units. "Let $a$ be the number of adult tickets"
- Find the two facts. Usually a total of things and a total of value, or two situations of the same kind
- Write one equation per fact
- Choose a method. Coefficient of $1$ means substitution; a matching or opposite pair means elimination
- Solve, then answer the question in a sentence
Step 5 is not decoration. A question asking "how many children" is not answered by $(20, 20)$.
Example 1: Count and Value
Here is the whole method on one problem, worked all the way through.
Adult tickets are $$8$, child tickets $$5$. Forty tickets sold, $$260$ taken. Let $a$ be adult tickets and $c$ be child tickets.
- Write the equation counting TICKETS a + c = 40
- Write the equation counting DOLLARS 8a + 5c = 260
- The first has coefficients of 1, so isolate c in it c = 40 - a
- Substitute into the money equation and simplify to the a term and a number 3a + 200 = 260
- Solve for a. How many adult tickets? 20
- Using c = 40 - a, how many child tickets? Write the subtraction you do 40 - 20