Read and write two-variable linear equations

Your feedback matters:

Lesson progressPractice problems 0/4

Difficulty: Intermediate
Estimated time: 36 minutes
Domains: Algebra
Techniques: Two-variable-constraintsStandard-formCoefficient-interpretationOrdered-pair-solutionsIntercept-relationships

What you’ll learn

Tell when one equation ties two quantities together.
Write and read standard form, $Ax+By=C$ .
Check whether a pair, a table row or a point on a graph is a solution.
Find one quantity when you know the other.
Connect ratios of the coefficients to the slope and the intercepts.

Prerequisites

Why this matters on the SAT

Two quantities, one total

Lots of SAT questions describe two kinds of things that have to add up to one total: two kinds of items, two distances, two amounts of money. One equation ties them together. It doesn't have just one answer, though. It describes every pair that fits the total. You might be asked to write that equation, say what one of its numbers means, test a pair, or find one amount when you're given the other.

Solution to the example

The letter $p$ counts premium posters, so $11$ is measured in dollars per premium poster:

\frac{11\text{ dollars}}{1\text{ premium poster}} \cdot p\text{ premium posters} = 11p\text{ dollars}.

So $11$ is the price of one premium poster, and the answer is A. Choice C is the trap: it describes the term $11p$ , the price of all $p$ premium posters together. Keep that split in mind whenever you read one of these equations: the coefficient is for one, and the term is for all of them.

SAT example

A print shop charges $7 for each standard poster and $11 for each premium poster. The equation

7s+11p=385

represents an order of $s$ standard posters and $p$ premium posters that costs $385 in total. Which choice is the best interpretation of $11$ in this context?

A
The price, in dollars, of each premium poster
B
The number of premium posters in the order
C
The total price, in dollars, of all premium posters
D
The difference, in dollars, between the prices of the two poster types

Recognize one constraint on two quantities

Take

4x+6y=48.

Each letter appears to the first power, times a number, and everything else is a plain number. That's what makes it a two-variable linear equation. In one of these, the letters are never multiplied together, like $xy$ , raised to a power, like $x^2$ , put under a root, like $\sqrt{x}$ , or put in a denominator, like $\frac{6}{y}$ .

This equation doesn't pin down one pair. $(0,8)$ , $(3,6)$ and $(12,0)$ all make it true, and so do plenty of others. In a word problem, an equation like this works as a constraint: a rule the two quantities have to follow together. It tells you which combinations are allowed, and neither letter is the input or the output.

A few questions look like this but need a different tool:

If one letter is the input and the other is the output, like a cost that depends on hours worked, the question wants a function rule. You'll find that in Write and match linear functions.
If you get two separate facts that must both be true, like a total cost and a total number of items, the two equations together can pin down one pair. That's a system, in Solve linear systems strategically.
If the story says at most, at least, less than or greater than, the solutions usually fill a region instead of a line. That's in Graph and model two-variable inequalities.
If the line has to be parallel or perpendicular to another line, you'll want Work with parallel and perpendicular lines.

Check your understanding:

A fundraiser earns $5 for each notebook and $8 for each planner, and $340 in total. Is that one constraint or a system? What can the two variables stand for?

Read standard form by following the units

Look back at $7s+11p=385$ and $4x+6y=48$ . They have the same shape: a number times one letter, plus a number times the other letter, equals a number. Put letters in place of those numbers and you get

Ax+By=C.

This shape is called standard form. $A$ , $B$ and $C$ are constants, and $A$ and $B$ can't both be zero.

In a word problem, each number has a job, and the units tell you what it is. Say a worker wraps $x$ small packages and $y$ large packages. A small package takes $4$ minutes, a large one takes $6$ minutes, and the whole job takes $120$ minutes:

4x+6y=120.

$4$ is the minutes for one small package.
$4x$ is the minutes for all the small packages.
$6$ is the minutes for one large package.
$6y$ is the minutes for all the large packages.
$120$ is the total minutes for both kinds together.

It's the poster split again, and the units show it. $4$ is in minutes per package, but $4x$ is in minutes, the same unit as the $120$ . A coefficient is usually a rate like this: a price per item, minutes per task, distance per trip. Multiply it by its letter and you get one part of the total. So in a story, standard form reads as two parts that add up to one fixed total:

\underbrace{Ax}_{\text{first contribution}} + \underbrace{By}_{\text{second contribution}} = \underbrace{C}_{\text{fixed total}}.

Try it yourself:

The equation $12h+8m=240$ gives the total minutes spent making $h$ handmade items and $m$ machine-made items. What do $12$ , $12h$ and $240$ each mean, and in what units? Decide before you read on.

Here's how it reads. $12$ is the minutes for one handmade item. $12h$ is the minutes spent on all the handmade items. And $240$ is the total minutes spent on both kinds.

One line, many standard forms

What happens if you divide every term of $4x+6y=120$ by $2$ ? You get

2x+3y=60.

Every pair that worked before still works. Try $(15,10)$ : $4(15)+6(10)=60+60=120$ , and $2(15)+3(10)=30+30=60$ . Multiplying or dividing every term by the same nonzero number never changes which pairs work. So the two equations have the same line, the same slope, the same intercepts and the same ratios between their coefficients. They describe the same constraint.

This next part is a little tricky: the numbers changed, so what they mean changed too. The new $2$ doesn't mean a small package now takes $2$ minutes. It still takes $4$ . Think of $2x+3y=60$ as counting time in 2-minute blocks instead of minutes. A small package takes $4$ minutes, which is $2$ blocks. A large one takes $6$ minutes, which is $3$ blocks. The whole job is $120$ minutes, which is $60$ blocks. Every number in the new equation is in blocks, not minutes.

Common mistake:

It’s tempting to read $A$ as the slope, or $C$ as an intercept, straight from $Ax+By=C$ . The form hides them. For example, $2x+3y=12$ has neither slope $2$ nor y-intercept $12$ : its slope is $-\frac23$ and it crosses the y-axis at $4$ . To find them, solve for one variable, or set the other one to zero.

Treat each solution as a complete pair

A solution is an ordered pair $(x,y)$ that makes the equation true. Keep the two numbers in order, and plug in both of them.

Take $4x+6y=48$ again, and try $(3,6)$ :

4(3)+6(6)=12+36=48.

That's true, so $(3,6)$ is a solution. Now flip it to $(6,3)$ :

4(6)+6(3)=24+18=42.

$42$ isn't $48$ , so the flipped pair is a different pair, and it isn't a solution. Order matters.

Every row of this table is one more solution pair.

A few solutions of $4x+6y=48$

$x$	$y$
$0$	$8$
$3$	$6$
$6$	$4$
$9$	$2$
$12$	$0$

The graph holds every real-number solution, not only these five rows. A word problem can narrow that down. If $x$ and $y$ count packages, negative numbers and half packages usually don't make sense, even though those points sit on the line.

When you know one quantity, plug it in and solve for the other, with the same steps you used in Solve linear equations. If $x=6$ ,

\begin{aligned} 4(6)+6y&=48\\[1.4em] 24+6y&=48\\[1.4em] 6y&=24\\[1.4em] y&=4. \end{aligned}

So the pair is $(6,4)$ , the third row of the table.

Check your understanding:

Does $(5,7)$ satisfy $3x+5y=50$ ? Check by plugging in, then give the value of $y$ that goes with $x=5$ .

Connect coefficients, slope, and intercepts

Standard form keeps the slope and the intercepts out of plain sight, but a little algebra brings them out. Let's use $4x+6y=48$ , the equation from the table.

Find the intercepts

At an intercept, the other coordinate is zero. That's the zero-coordinate rule from Understand intercepts and starting values.

For the x-intercept, set $y=0$ . That leaves $4x=48$ , so $x=12$ . For the y-intercept, set $x=0$ . That leaves $6y=48$ , so $y=8$ . The intercepts are $(12,0)$ and $(0,8)$ , the first and last rows of the table.

The same two moves work on any equation in standard form. For the x-intercept, set $y=0$ . If $A\ne0$ ,

Ax=C \quad\Longrightarrow\quad x=\frac{C}{A}.

For the y-intercept, set $x=0$ . If $B\ne0$ ,

By=C \quad\Longrightarrow\quad y=\frac{C}{B}.

So when neither coefficient is zero, the intercepts are

\left(\frac{C}{A},0\right) \qquad\text{and}\qquad \left(0,\frac{C}{B}\right).

In a story where $x$ and $y$ count two kinds of items, these points are often the all-x and all-y cases, where only one kind is bought or sold, as long as the story allows that.

Find the slope

To see the slope, solve for $y$ . With $4x+6y=48$ , subtract $4x$ and divide by $6$ :

\begin{aligned} 6y&=-4x+48\\[1.4em] y&=-\frac{2}{3}x+8. \end{aligned}

The slope is $-\frac23$ , which is $-\frac46$ : the negative of $4$ over $6$ . The same steps work with letters, as long as $B\ne0$ :

\begin{aligned} Ax+By&=C\\[1.4em] By&=-Ax+C\\[1.4em] y&=-\frac{A}{B}x+\frac{C}{B}. \end{aligned}

So the slope is $-\frac{A}{B}$ , not $A$ or $B$ on its own. That's the rate of change from slope and rate of change, written with the coefficients. And the y-intercept, $\frac{C}{B}$ , matches what the zero-coordinate rule gave.

You can also line up both intercepts in one equation. Divide every term of $4x+6y=48$ by $48$ . That's the same scaling move as before, so the line doesn't change:

\frac{x}{12}+\frac{y}{8}=1.

The numbers under $x$ and $y$ are the intercepts, $12$ and $8$ . With letters, when $A$ , $B$ and $C$ are all nonzero, dividing by $C$ gives

\frac{x}{C/A}+\frac{y}{C/B}=1,

and the denominators are the x- and y-intercepts.

Graph it to see the whole line

The current College Board graphing calculator takes standard form as it is. Type 10x+16y=320; there's no need to solve for $y$ first. Click the line, and you'll see where it crosses the axes: $(32,0)$ and $(0,20)$ .

That's what the graph is good for: it shows every real solution and both intercepts at once.

Check your understanding:

For $6x+15y=90$ , find both intercepts and the slope. Then, if $x$ and $y$ count two kinds of tickets that bring in $90 in total, what does the x-intercept mean?

Calculator loads as you approach

Every point on the line is a solution. The two marked points are the all-x and all-y cases.

Let the question pick the method, not the fact that a calculator is open.

Start by hand when…

you have to build the equation, like turning $7 per standard poster and $11 per premium poster into $7s+11p=385$ .
you’re asked what a number or a pair means, like the $11$ in $7s+11p=385$ or the pair $(3,6)$ .
one quantity is given and what’s left is short, like $4(6)+6y=48$ .
you’re matching an equation to a table with clean numbers, like checking that the row $(9,2)$ fits $4x+6y=48$ .
the coefficients or intercepts involve a letter like $k$ , so the relationship has to stay exact.

Graph the equation as written when…

you want to see the whole solution line, not just one pair.
you need both intercepts, or you’re comparing several points.
finding an intercept means an awkward decimal or fraction division, like $27.3\div4.2$ .

When you do graph, type the equation just as it’s written, with no rearranging, and click the crossing on the axis the question asks about.

A graph can’t tell you whether negative or fractional values, or any others the story rules out, make sense. That part is always your call.

Example: Read one constraint from end to end

Worked example

A school theater sells student tickets for $12 each and adult tickets for $18 each. The theater collects $504 in total from $s$ student tickets and $a$ adult tickets. If the theater sells $24$ student tickets, how many adult tickets does it sell?

A
$8$
B
$12$
C
$24$
D
$28$

Step 1

Name what each letter counts

The letter $s$ counts student tickets, so $12s$ is the money from student tickets. The letter $a$ counts adult tickets, so $18a$ is the money from adult tickets.

Both terms are in dollars.

Step 2

Write the total

The two amounts add up to the $504 total:

12s+18a=504.

This one equation covers every mix of student and adult tickets that brings in $504.

Step 3

Plug in what you know

The question says $s=24$ :

\begin{aligned} 12(24)+18a&=504\\[1.4em] 288+18a&=504\\[1.4em] 18a&=216\\[1.4em] a&=12. \end{aligned}

Step 4

Answer, then check the pair

The theater sells $12$ adult tickets, so the answer is B. As a pair, that's $(24,12)$ , since the letters go in the order $(s,a)$ .

Check the total:

12(24)+18(12)=288+216=504.

Check your understanding:

What are the two intercepts of $12s+18a=504$ , and what does each one mean for the tickets?

Common mistake:

Don’t treat $(24,12)$ as the only solution. That mixes up one constraint with a system. The extra fact $s=24$ picks one point off the line. Without it, lots of ticket mixes make $12s+18a=504$ true, like $(27,10)$ or $(42,0)$ .

Practice problems

Your turn. Keep track of what each letter counts, and answer exactly what the question asks.

Write the time equation

Practice problem

A museum prepares $x$ adult audio guides and $y$ student audio guides. Preparing each adult guide requires $6$ minutes, and preparing each student guide requires $9$ minutes. The museum spends $270$ minutes preparing the guides. Which equation represents this constraint?

Calculator loads as you approach

Build the equation from the units by hand. Graph it here if you want to see its line.

Read a scaled equation

Practice problem

A pottery studio uses $12$ pounds of clay for each large planter and $8$ pounds of clay for each small planter. If $l$ is the number of large planters and $s$ is the number of small planters, the studio’s clay constraint is

12l+8s=240.

Dividing every term by $4$ gives

3l+2s=60.

Which statement correctly interprets the rewritten equation?

Calculator loads as you approach

Work out the units by hand. To check, use

x

for

l

and

y

for

s

, graph

12x+8y=240

and

3x+2y=60

, and see that they draw the same line.

Click an awkward intercept

Practice problem

The graph of

4.6x+7.3y=52.9

has an x-intercept at $(a,0)$ . What is the value of $a$ ?

Calculator loads as you approach

Type the equation as written, then click where the line crosses the x-axis.

Find k from both intercepts

Practice problem

The graph of

(k+2)x+(k-1)y=6k

has an x-intercept at $(a,0)$ and a y-intercept at $(0,b)$ , where $k\ne-2$ and $k\ne1$ . If $a+b=12$ , what is the value of $k$ ?

Your answer

Calculator loads as you approach

Work out the intercepts exactly, by hand. Graph here only to check your value of k.

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

One two-variable linear equation usually has many solution pairs, not just one.
In $Ax+By=C$ , the terms $Ax$ and $By$ are two parts that add up to one fixed total.
The coefficient is for one, and the term is for all of them. The units tell you which is which.
Multiplying or dividing every term by the same nonzero number keeps the solutions, graph, slope, intercepts and coefficient ratios. The coefficients change, though, and so does what they mean, like minutes turning into 2-minute blocks.
A pair is a solution only when both numbers, in the right order, make the equation true.
If you know one quantity, plug it in and solve for the other.
To find an intercept, set the other letter to zero. When the coefficients involved aren't zero, the intercepts are $\left(\frac{C}{A},0\right)$ and $\left(0,\frac{C}{B}\right)$ , and the slope is $-\frac{A}{B}$ .
Work by hand when you're building an equation, saying what a number means, checking a pair or a table row with clean numbers, or finding one value from the other. Keep letters like $k$ exact.
When an intercept means awkward division, type the equation into Desmos as written, click the line and click the crossing on the axis you need.

Related lessons

Go back to Understand intercepts and starting values for more practice reading the points where one coordinate is zero, or try Write and match linear functions when one variable is set up as the input and the other as the output.

Next lesson

Work with parallel and perpendicular lines

Use equal and negative-reciprocal slopes to build or analyze related lines.

Start next lesson

Practice

Practice this lesson

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

Start practice