Linear Equation Word Problems
Translate word problems into equations, solve, and interpret the answer in context. Cover number problems, geometry problems, mixture problems, and rate/distance/time problems.
A lesson of Algebra 1, in Solving Linear Equations, about 40 minutes.
By the end of this lesson, you will be able to model and solve the five word-problem types that come up most often, number relationships, rate and distance, mixtures, geometry and break-even, and interpret each answer back in the words of the problem.
The Equation Is the Easy Part
By now you can solve any linear equation put in front of you. Almost nobody loses points on that. What people lose points on is the gap between a paragraph of English and the first line of algebra.
This lesson closes that gap five times over, because five setups cover most of what you will ever be asked. Once you recognize which one you are looking at, the algebra is work you have already done.
The Protocol, Every Time
- Read the whole problem before writing anything
- Define the variable: "Let $x$ be ...", with units
- Express the other unknown quantities in terms of it
- Write the equation that states the key relationship
- Solve it
- Interpret: answer the question that was asked, not just find $x$
- Verify against the original words
Step 6 is where good algebra loses points. If $x$ is the width and the question asked for the length, then $x$ is not the answer, it is halfway to the answer.
Example 1: Two Numbers with a Rule Between Them
Here is the whole method on one problem, worked all the way through.
The sum of two numbers is 48. One number is 4 times the other. Name the simpler quantity. Let $x$ be the smaller number, so the larger is described in terms of it.
- Write the larger number in terms of x 4x
- Write the equation saying the two numbers add to 48 x + 4x = 48
- Combine the like terms on the left 5x = 48
- Divide both sides by 5 x = 9.6
Worked Example 1: Consecutive Odd Integers
Three consecutive odd integers add to 57. Odd numbers are two apart, so once you name the smallest the other two follow. Let $n$ be the smallest.
Example 2: Two Things Moving
Every rate problem is built on one fact: $$d = rt$$ When two things move, write that for each of them separately, then say how the two are related.
Two trains start 360 miles apart and drive toward each other. One does 80 mph, the other 100 mph. They travel for the same length of time, so one letter covers both. Let $t$ be the hours until they meet.
- Write the distance the first train covers in that time 80t