Chapter 7 · Linear equations in one variable
Word Problems with Variables on Both Sides
Turn a story about two changing amounts into an equation with the variable on both sides. Solve it, check it, and say what the answer means.
Some questions compare two amounts that are both changing. Maya has 40 dollars and saves 12 dollars each week. Leo has 100 dollars and saves 7 dollars each week. Maya starts behind, but she saves faster. Will she catch up with Leo, and if so, when?
Both amounts depend on the same unknown number: how many weeks have passed. Before you write any symbols, follow the two amounts for a few weeks.
| Weeks passed | Maya's savings (dollars) | Leo's savings (dollars) | Leo's lead (dollars) |
|---|---|---|---|
Every row follows the same pattern, so you can describe all the weeks at once. Let stand for the number of weeks that have passed. A letter used this way is a variable: it stands for a number that can change or that you do not know yet. Here counts whole weeks, so the values it is allowed to take, called its domain, are , and so on.
After weeks, Maya has dollars and Leo has dollars. Each of these is an expression: numbers and variables joined by operations, with no equals sign. The 12 and the 7 are rates, in dollars per week: how much each amount changes for every 1 week that passes.
The last column of the table shows Leo's lead shrinking by dollars each week. Maya catches up when the lead reaches 0, which is when the two amounts are equal. You could keep adding rows, but an equation, a statement that two expressions have the same value, asks the question directly. A value of that makes the equation true is a solution. Both amounts depend on the number of weeks, so appears on both sides:
Key concept
Two amounts, one variable, one equation
When a story asks when two amounts are equal, and both depend on the same unknown number, write one expression for each amount and set the expressions equal.
- Define the variable: what it counts, its unit and its domain.
- Model each amount and write the equation.
- Solve by doing the same valid operation to both sides, which keeps the solutions the same: add or subtract the same quantity, or multiply or divide by the same number that is not zero. Never divide by zero.
- Check your value in the original equation and in the domain. Then answer in words.
If the variable drops out, a true statement makes the equation an identity: every value in the domain is a solution. A false statement makes it a contradiction: it has no solution.
Worked example
When do the savings match?
Maya has 40 dollars and saves 12 dollars each week. Leo has 100 dollars and saves 7 dollars each week. After how many weeks do they have the same amount, and how much does each of them have then?
Define the variable before you use it. Let w be the number of weeks that have passed, so its unit is weeks. It counts whole weeks, so its domain is 0, 1, 2, 3, and so on.
Write one expression for each person. Each starts with a one-time amount and then adds the same number of dollars every week, so the rate multiplies w. Both expressions are in dollars. The question asks when the amounts are the same, so set the expressions equal.
Subtract 7w from both sides so that w is on one side only. Subtracting the same quantity from both sides is a valid operation, so the solutions stay the same. Then simplify. On the left, 12w - 7w and 5w are equivalent expressions: they have the same value for every w in the domain, so you can swap one for the other. On the right, 7w - 7w is 0.
Subtract 40 from both sides to leave 5w alone. This is the same valid operation on both sides again, so the solutions still do not change.
Divide both sides by 5 to find w. Dividing both sides by the same number keeps the solutions the same only when that number is not zero, and 5 is not zero.
Check 12 in the original equation, not in a later line, so that a slip in any step would show up. Both sides equal 184, so the equation is true, and 12 is a whole number of weeks, so it is in the domain. It also fits the table: a lead of 60 dollars that shrinks by 5 dollars a week is gone after 12 weeks.
Answer
Maya and Leo have the same amount after weeks. At that point each of them has dollars.
Worked example
Two water tanks
Tank A holds 90 liters of water and drains steadily at 2.5 liters per minute. Tank B holds 18 liters and fills steadily at 3.5 liters per minute. The water flows for 30 minutes. When do the two tanks hold the same amount of water, and how much is that?
Let t be the number of minutes since the water started flowing. The flow is steady, so t does not have to be a whole number. Its domain is every number from 0 to 30.
Tank A loses water, so subtract 2.5 liters for each minute from its 90 liters. Tank B gains water, so add 3.5 liters for each minute to its 18 liters. Both expressions are in liters. Set them equal because the question asks when the amounts match.
Add 2.5t to both sides. The left side subtracts 2.5t, so adding 2.5t cancels it, and doing the same valid operation to both sides keeps the solutions the same.
Subtract 18 from both sides, the same valid operation again, to leave 6t alone.
Divide both sides by 6. This keeps the solutions the same because 6 is not zero.
Check 12 in the original equation. Both sides give 60 liters, so the equation is true, and 12 is between 0 and 30, so it is in the domain.
Answer
The tanks hold the same amount of water minutes after the start. At that moment each tank holds exactly liters.
In both stories so far, the two rates were different. The gap between the amounts changed steadily and reached zero exactly once. When two amounts like these change at the same rate, the gap never changes. Then the variable drops out as you solve, and you meet one of the special cases from the key idea: a contradiction when the starting amounts are different, or an identity when the starting amounts are the same too.
Worked example
A gap that never closes
Print shop A charges a setup fee of 15 dollars plus 0.75 dollars per poster. Print shop B charges a setup fee of 9 dollars plus 0.75 dollars per poster. For how many posters do the two shops charge the same total?
Let p be the number of posters in the order. You order at least one whole poster, so the domain is 1, 2, 3, and so on. Each total, in dollars, is the one-time setup fee plus 0.75 dollars for each poster. Set the two totals equal.
Subtract 0.75p from both sides. This is the same valid operation on both sides, so the solutions do not change, but this time p drops out of the equation.
Read what is left. The statement 15 = 9 is false for every value of p, so the equation is a contradiction and has no solution.
Make sense of this in the story by testing 10 posters. Shop A charges 22.50 dollars and shop B charges 16.50 dollars. Both shops add the same 0.75 dollars per poster, so shop A always costs exactly 6 dollars more, which is the difference between the setup fees.
Answer
No number of posters gives the same total, because the equation has no solution. Shop A always charges exactly dollars more than shop B.
Takeaway. A word problem with the variable on both sides compares two amounts that depend on the same unknown number. Define the variable with its unit and domain, write one expression for each amount, and set the expressions equal. Solve by doing the same valid operation to both sides, never dividing by zero. Check your value in the original equation and in the domain, then answer the question the story asked. If the variable drops out, a false statement means the amounts are never equal, and a true statement means they are equal for every value in the domain.