Model and interpret one-variable equations

Your feedback matters:

Lesson progressPractice problems 0/3

Difficulty: Beginner
Estimated time: 30 minutes
Domains: Algebra
Techniques: Contextual-equationsVariable-definitionTerm-interpretationUnit-checking

What you’ll learn

Pick one unknown and say exactly what it counts, with its unit.
Turn rates, totals and comparisons into one linear equation.
Explain what a constant, a coefficient, a term and the solution mean in the story.
Decide whether to solve the finished equation by hand or enter it intact in Desmos.
Check a model by matching units and testing its answer.
Use the same idea for work-rate and mixture problems.

Prerequisites

Solve linear equations0%
Use equivalent steps to solve the one-variable equation after you build it.

Why this matters on the SAT

Turn the story into one equation

Lots of SAT Math questions hand you a story instead of an equation. Your first job isn't to calculate. It's to figure out what each number does, pick one unknown, and connect the pieces with an equal sign. In short: give every number a job.

You'll want to be comfortable with signed numbers and simple fractions. The solving steps themselves come from Solve linear equations.

Solution to the example

Start with the part that repeats. Each class costs $16, so $c$ classes cost $16c$ dollars. The $28 fee is paid once, so it's added once:

\underbrace{16c}_{\text{class charges}} + \underbrace{28}_{\text{one-time fee}} = \underbrace{140}_{\text{total paid}}.

That's choice C. Choice A swaps the jobs: $28c$ would mean $28 for every class. Solving C gives $c=7$ , so the student went to $7$ classes. Notice where the real work happened: in deciding what each number does, before any algebra.

Try it yourself:

When a story asks for one unknown amount, start by saying what your letter stands for. Then build each piece of the equation from the numbers and their units. Solving comes last.

SAT example

An art studio charges a one-time registration fee of $28 and $16 for each class a student attends. A student paid a total of $140 for the registration fee and $c$ classes. Which equation represents this situation?

A
$28c+16=140$
B
$16c-28=140$
C
$16c+28=140$
D
$16(c+28)=140$

When one unknown is enough

A story can mention several amounts you don't know and still need only one unknown. That works when, once you pick one of them, the others follow from it.

You'll often see one of these setups:

a fixed amount plus a repeated amount makes a total
two related groups add up to a known total
one amount is more or less than another
two expressions describe the same amount
part of a mixture, or part of a job, has to hit a target

Here's how the second one works. Say $x$ of $30$ lab stations are chemistry stations. The rest aren't a second unknown. Once you know $x$ , the number of other stations is

30-x.

So both groups fit into one equation in $x$ .

Some look-alike questions need a different approach:

If the equation is already written and you only need to solve it, see Solve linear equations.
If two amounts can each change on their own, so you need two separate facts to pin them down, you need a system. See Solve linear systems strategically.
If the story says at least 20, fewer than 8 or no more than 50, it describes a range, not one exact value. That calls for Solve one-variable inequalities.
If the question asks whether an equation has one solution, none or infinitely many, see Count solutions and determine constants.

Check your understanding:

A shipping center packs $46$ boxes in total. If $x$ is the number of large boxes, how would you write the number of small boxes? And why is this still a one-variable model?

Build the equation piece by piece

You can build almost any of these equations in four moves.

1. Say exactly what your letter means

Write down what it counts or measures, and include the unit:

Let $h$ be the number of hours the equipment is rented.

That one line keeps $h$ from quietly changing meaning halfway through the problem.

2. Write every other amount with that letter

A rate times an amount gives that piece of the total:

\text{rate}\times\text{amount}=\text{contribution}.

At $24 per hour for $h$ hours, the hourly charge is $24h$ dollars. The $24$ carries the rate, and $h$ carries the hours.

When two groups share a total, write the second group as the total minus the first, like the $30-x$ stations above.

3. Find the sentence that means "equals"

The equal sign should stand for a sentence in the story, such as

\text{fixed charge}+\text{hourly charges}=\text{total charge}

\text{contribution from group 1}+\text{contribution from group 2} = \text{known total}.

4. Check the units before you solve

Anything you add has to be in the same unit. In

24h+65=353,

$24h$ , $65$ and $353$ are all dollars, because

\frac{\text{dollars}}{\text{hour}}\times\text{hours} = \text{dollars}.

If one piece counted hours and another counted dollars, you couldn't add them as parts of one total. A mismatch like that tells you the model is wrong before you do any algebra.

Try it yourself:

A regular ticket costs $p$ dollars, and a student ticket costs $6 less. Write the student price in terms of $p$ . Then write the total cost of $3$ regular tickets and $2$ student tickets, using only $p$ . Try it before you read on.

The student ticket costs $p-6$ dollars, so the total is

3p+2(p-6).

Common mistake:

It’s tempting to write “$6 less than $p$ ” as $6-p$ , in the order you read it. That flips the comparison. With $p=20$ , $6-p$ gives a student price of $-14$ dollars, which can’t be right. Start from $p$ and take $6$ away: $p-6$ . With $p=20$ , that gives a sensible $14.

What each part means

Say a kayak rental is modeled by

24r+65=353,

where $r$ is the number of hours rented. Each part has its own job:

The variable $r$ is the number of rental hours.
The coefficient $24$ , the number multiplying $r$ , is the hourly rate in dollars per hour.
The term $24r$ is the whole hourly charge, in dollars.
The constant term $65$ is the one-time fee, in dollars.
The right side, $353$ , is the total charge, in dollars.
The solution is $r=12$ , so the kayak was rented for $12$ hours.

That last line matters more than it looks. The answer means whatever the letter means. "The solution is $12$ " doesn't say $12$ of what. Here it's 12 rental hours, not $12.

Check your understanding:

In $24r+65=353$ , what does $24r$ stand for? How are its units different from the units of $24$ ?

Common mistake:

Calling $24r$ the hourly rate mixes up a term with its coefficient. The rate is $24$ dollars per hour, and $24r$ is the total charge after $r$ hours. If you’re not sure which is which, divide the term’s units by the variable’s units: dollars divided by hours gives dollars per hour, the rate.

Model first, then pick how to solve

In these questions, the hard part is almost always the model, not the arithmetic after it.

Building the model is your job

If the question asks which equation fits, match each number to its job, like the one-time $28 fee in $16c+28=140$ .
If it asks what a number, term or solution means, go back to what the letter stands for, like $r$ counting rental hours.
If you’re not sure how two amounts combine, check the units: dollars per hour times hours gives dollars.
If two groups share a total, write the second one as the total minus the first, like $30-x$ .

Once the equation is right, pick how to solve

If it takes one or two quick exact steps, like $16c+28=140$ , solve it by hand.
If decimals, fractions or distributing would drag the arithmetic out, like $0.12(18)+0.30x=0.20(18+x)$ , enter the whole equation intact in Desmos, exactly as written.
If plugging in the answer choices is truly quicker than solving, do that instead.

Desmos can solve an equation, but it can’t tell you what the numbers mean. That part is always yours.

Want more practice solving equations in Desmos? Try Solve one-variable equations.

Entering the equation intact means you don't distribute or clear decimals first. Once the model is right, those extra hand steps are just more places for a slip to sneak in.

Work rates and mixtures

These stories look harder, but the idea is the same: pick one unknown, write each piece with it, and set matching totals equal.

Work-rate problems

If one machine finishes a job in $9$ hours, it does $\frac19$ of the job each hour. If a second machine takes $18$ hours for the same job, it does $\frac1{18}$ of the job each hour.

Say they work together for $x$ hours. Add up the part of the job each one finishes:

\underbrace{\frac{x}{9}}_{\text{first machine's part}} + \underbrace{\frac{x}{18}}_{\text{second machine's part}} = \underbrace{1}_{\text{one complete job}}.

The parts add because both machines are working on the same job. The times, $9$ and $18$ , don't add. A phrase worth remembering: add the work, not the times.

Once the equation shows the parts of the job, enter it intact in Desmos:

x/9+x/18=1

Desmos draws it as a vertical line. Click that line and you'll see $x=6$ . That makes sense: with help, the faster machine should finish sooner than its $9$ hours alone.

Check your understanding:

A pump can fill a tank in $8$ hours, and a second pump can fill the same tank in $12$ hours. If they work together for $x$ hours, what equation says they fill one full tank?

Mixture problems

In a mixture, keep track of how much of the ingredient there is, like the amount of salt:

\text{ingredient amount} = \text{concentration}\times\text{solution volume}.

A quick percent refresher: percent means per 100, so divide by $100$ to turn a percent into a decimal:

12\%=\frac{12}{100}=0.12.

So $12\%$ of $18$ liters is $0.12(18)$ liters.

Here's the tricky part. When you pour in $x$ more liters, the total volume grows too. Start with $18$ liters, add $x$ liters, and you have $18+x$ liters, not $18$ .

So a mixture equation reads like this:

\text{ingredient already present} + \text{ingredient added} = \text{ingredient in the final mixture}.

Say an $18$ -liter solution that's $12\%$ salt by volume gets $x$ liters of a $30\%$ salt solution, and the result is $20\%$ salt by volume. The model is

0.12(18)+0.30x=0.20(18+x).

Every term is liters of salt. The right side uses $18+x$ because the $20\%$ is a percent of the final mixture, not the starting one.

This one has decimals on both sides, so enter it intact on the next line of the same calculator:

0.12(18)+0.30x=0.20(18+x)

Click its vertical line to get $x=14.4$ . So $14.4$ liters of the $30\%$ solution were added.

Common mistake:

The classic trap is writing $0.20(18)$ on the right. That takes $20\%$ of the old $18$ liters, but after you pour in $x$ more liters, the mixture is bigger. Write the final volume first, $18+x$ , and then take $20\%$ of it. A last check: every term should be liters of salt.

Calculator loads as you approach

Each finished equation has its own line. Click a line’s vertical graph to read its solution:

x=6

for the work-rate model and

x=14.4

for the mixture.

Practice problems

Your turn. For each one, say what the unknown means, build or read the equation, and give your answer in the units the question asks for.

Model two related groups

Practice problem

A community center prepares $26$ activity kits. Each art kit requires $8$ markers, and each science kit requires $5$ markers. The center uses $154$ markers in total. How many art kits does the center prepare?

Your answer

Calculator loads as you approach

Write both marker counts first. Then solve or check the finished equation here if it helps.

What does the solution mean?

Practice problem

A print shop charges $18 for each banner plus a one-time setup fee of $54. The equation

18b+54=342

models an order that costs $342 in total, where $b$ is the number of banners ordered. Which choice is the best interpretation of the solution to the equation?

Calculator loads as you approach

Solve or check the equation here if it helps.

Combine two work rates

Practice problem

One machine can package an order in $14$ hours. A second machine can package the same order in $21$ hours. If both machines work together at their constant rates, how many hours will it take them to package one order?

Your answer

Calculator loads as you approach

Write the equation for the parts of the order first. Then enter it intact here and read the time.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Say what your one unknown means, with its unit, before you build anything.
Write every related amount with that same letter, like $30-x$ for the other group.
A rate times an amount gives one piece of a total. Set matching totals equal.
Units tell a coefficient from its term: $24$ is dollars per hour, $24r$ is dollars.
Solve short, clean equations by hand. Enter messy finished ones intact in Desmos.
The answer means whatever the letter means. Check it against the story.
For work rates, add the work, not the times. For mixtures, track the ingredient and the new total volume.

Next lesson

Count solutions and determine constants

Decide when a linear equation has one, no, or infinitely many solutions and find constants that create each case.

Start next lesson

Practice

Practice this lesson

1,237 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

Start practice

What you’ll learn

Prerequisites

Turn the story into one equation

When one unknown is enough

Build the equation piece by piece

1. Say exactly what your letter means

2. Write every other amount with that letter

3. Find the sentence that means "equals"

4. Check the units before you solve

What each part means

Model first, then pick how to solve

Building the model is your job

Once the equation is right, pick how to solve

Example: Model two related groups

Write both counts with x

Count each group’s trays

Set them equal to the total

Solve, then check it against the story

Work rates and mixtures

Work-rate problems

Mixture problems

Practice problems

Model two related groups

What does the solution mean?

Combine two work rates

3 practice examples left

Quick recap

Count solutions and determine constants

Practice this lesson