You graph y = 4x - 9 and y = x + 6, click the crossing, and Desmos labels it (5, 11). What is the solution to 4x - 9 = x + 6?

5 — The solution is the x-coordinate. 11 is what both sides are worth when x is 5, which is a good check that you graphed the right thing but is not what was asked. Expect to see it among the answer choices.

Write out the keys you press, in order, to get y = x/4 - 3 onto a Desmos line.

y = x, then /, then 4, then the right-arrow key, then - 3. — Without the right arrow the - 3 lands in the denominator, so you graph x over (4 - 3), which is just y = x. It looks perfectly tidy on screen, and that is what makes this mistake expensive: nothing tells you it is wrong.

A pool holds 200 litres and drains at 4 litres per minute. A second pool holds 80 litres and fills at 6 litres per minute. You graph y = -4x + 200 and y = 6x + 80, and the crossing is at (12, 152). The question asks when the two pools hold the same amount. Which number do you give?

12 — You set x as minutes, so "when" is the x-coordinate: 12 minutes. Had the question asked how much each pool holds, the answer would have been 152 litres. Same graph, same click, different coordinate. The wording decides, not the picture.

In the equation (k/5)x + 2 = 4x - 7, k is a constant and the equation has no solution. What is the value of k?

20 — No solution means the two sides are parallel lines, so their slopes are equal: k/5 = 4, which gives k = 20. Answering 4 means you stopped at the slope. If you want the arithmetic done for you, type 5*4 on a bare Desmos line.

Linear Equations in One Variable on the Digital SAT: the Desmos Method

An equation asks where two expressions are worth the same. Graph the left side and the right side as two separate lines, click where they cross, and read off the coordinate the question actually asked for.

12 min read·Algebra·65 practice problems in DesPrep

How often it comes up

Roughly 2-4 of the 44 math questions are a linear equation in one variable, spread across both modules and clustered near the front of each. Algebra as a whole is about a third of the section, and the two-graph move you learn here gets reused on systems, on inequalities and on most rate word problems, so it earns a good deal more than those 2-4 questions.

Why this is on the test

Algebra is the largest domain on the digital SAT, and a one-variable linear equation is the cheapest question in it. The College Board keeps asking it because it is the floor of the whole test — plenty of harder questions end with a linear equation you still have to solve, so being shaky here leaks points in places that are not even labeled algebra. Every question is worth the same, so a mark dropped on 4(x + 5) = -12 costs exactly as much as a mark dropped on the hardest question in Module 2. It costs you twice, in fact: you lose the mark, and you lose the ninety seconds you spent hunting for the sign error, and that time comes straight out of the questions at the end. Module 1 also decides which Module 2 you are given, and these questions sit early in Module 1 where they are meant to be quick. Solving one by hand takes most students sixty to ninety seconds and carries a real chance of a slip. Graphing it takes about twenty, and a graph cannot make a sign error.

The Desmos playbook for linear equations in one variable

When you see the question on the left, type the middle line into Desmos and read the number on the right. This is the whole skill, in 6 moves.

When you see	Type in Desmos	Read off
x appears on both sides, like 3x - 11 = 9x - 59.	y = 3x - 11, then Enter, then y = 9x - 59.	Click the crossing. The x-coordinate is the solution. The y-coordinate will be one of the wrong choices.
One side is just a number, with or without a bracket, like 4(x + 5) = -12.	y = 4(x + 5), then Enter, then y = -12. Leave the bracket as printed.	A flat line cutting a slanted one, once. Read the x-coordinate.
There is a fraction anywhere in it, like x/5 + 7 = 2.	y = x, then /, then 5, then the RIGHT ARROW key, then + 7. Enter, then y = 2.	The x-coordinate of the crossing, but first check on screen that the + 7 is outside the fraction bar.
Two quantities changing at steady rates, and the question names a unit.	y = (rate)x + (starting amount) for each one, with a negative rate for anything going down.	Click the crossing, then take the coordinate whose units the question named. "How much" is usually y; "when" or "after how many minutes" is usually x.
It says the equation has no solution and asks for the value of k, with k inside the slope.	Set the two slopes equal, then type the arithmetic on a bare line, such as 6*3 for k/6 = 3.	The number Desmos prints under the line is k. It is not the slope you matched against.
It says the equation has infinitely many solutions, with the right side written px + q.	Distribute the bracket, then type the total on a bare line, such as 33+31.	The printed number is p + q. Re-read the question to check it wanted p + q and not p or q on its own.

Before you start, you should be able to

Type an expression on a Desmos expression line and see it draw.
Tell the left side of an equation from the right side.
Read an ordered pair off a graph and say which number is x and which is y.
Click a point in Desmos to make its coordinates appear.
Turn a short phrase into an expression, like "twice the largest" into 2(x + 4).

01An equation is a question about two graphs

5x + 7 = 3x + 11 looks like an instruction to start doing things to both sides. It is not. It is a question: at what value of x are 5x + 7 and 3x + 11 worth the same number?

Give each side its own graph. Type y = 5x + 7 and you get a line whose height at any x is the value of the left side. Type y = 3x + 11 and you get a second line whose height is the value of the right side. Everywhere the two lines sit at different heights, the two sides disagree. At the one place they touch, they agree. That x is the solution.

That is the entire skill. Everything else in this lesson is about typing it in correctly and reading the right number off the point.

02The three moves, every time

Three moves, in this order, on every question of this shape.

One: the left side on its own line, typed as y = (left side). Two: the right side on the next line, typed as y = (right side). Three: click the point where the two graphs cross, and read the coordinate the question asked for.

Copy each side in exactly as printed. Do not simplify first, do not move terms across, do not clear a bracket. Every one of those is a chance to lose a sign, and Desmos does not need any of them.

Put each side on its own line. Do not type the whole equation onto one line and hope. The step checklist in practice is watching for the two sides separately, and the two-line picture is the only version that still works when the question asks for something other than x, which happens more often than you would think.

If you cannot see a crossing, the window is too small — the method is fine. Zoom out with the minus button at the right of the graph until the two lines meet.

The rest of this lesson is in DesPrep

Linear Equations in One Variable continues with 4 more sections, and a worked example on every one of them:

Fractions, and the one place Desmos will quietly disagree with you
Read the coordinate the question asked for
Word problems: you build the two lines, Desmos does the rest
The two questions that are not about the crossing

Then 65 practice problems on this skill alone, solved inside a live Desmos calculator with step feedback and an expert replay on every one.

See plans — from $9.99/mo Or start with the 2 free skills

Mistakes that cost marks on linear equations in one variable

Reading the y-coordinate of the crossing instead of the x-coordinate.

The point gives you two numbers and only one was asked for. On 5x + 7 = 3x + 11 the crossing is (2, 17), and 17 is a lettered choice. It is what both sides are worth at the solution, which is a real and useful number, just not the answer. This is the most common way a correct graph turns into a wrong mark.

FixSay "I want x" before you click. On anything phrased as the solution, or the value of x, take the first number in the bracket and move on.

Reading the x-coordinate when the question asked for a quantity.

In the tank questions x is minutes and y is litres. The crossing at (12, 106) offers you 12, which is the time, not the amount, and both numbers are in the choices. The habit built on the easy questions fires on a question where it is wrong.

FixWrite down what x means and what y means before you type. Then match the unit in the question, litres or dollars or minutes, to one of them.

Typing y = x/2 + 4 straight through.

The / key drops the cursor into the denominator and leaves it there, so the + 4 goes under the bar. You have graphed x over 6. Desmos draws it happily, the crossing is clean, and the number you read is simply wrong.

FixSlash, denominator, right arrow, then the rest. Then glance at the screen: the fraction bar must sit under only what you meant to divide by.

Doing one step of the algebra and stopping.

On 4(x + 5) = -12, dividing by 4 gives x + 5 = -3 and the -3 gets written down. On x/2 + 4 = 6, subtracting 4 gives x/2 = 2 and the 2 gets written down. Both are answer choices, because a half-finished solution is exactly what the distractors are built from.

FixThis is the whole argument for graphing. A graph has no half-finished state. The crossing is either on the screen or it is not.

Losing a sign while rearranging by hand.

3x - 6 = -2x + 14 solves to 4, and -4 is sitting in the choices. Moving a negative term across the equals sign is the most reliable way to drop one of these marks, and you will not notice, because -4 looks like an answer.

FixType each side in exactly as printed and let the picture handle the signs. Even if you already solved it on paper, graph it; the check costs about ten seconds.

Answering the slope instead of k, or p instead of p + q.

In (k/6)x - 11 = 3x - 5 the slopes match when k/6 = 3, so k = 18. Answering 3 means you found the slope and called it k. The same shape appears on the other variant: you work out p = 9 and stop, when the question asked for p + q. Both of those wrong values are printed as choices.

FixUnderline what the question asks for before you start, and read it again after Desmos prints a number. These two variants test whether you finish, not whether you can find a slope.

Check yourself on linear equations in one variable

4 questions that are not in the practice bank. Work each one the Desmos way before you open the answer.

1
You graph y = 4x - 9 and y = x + 6, click the crossing, and Desmos labels it (5, 11). What is the solution to 4x - 9 = x + 6?
Show the answer
Answer: 5
The solution is the x-coordinate. 11 is what both sides are worth when x is 5, which is a good check that you graphed the right thing but is not what was asked. Expect to see it among the answer choices.
2
Write out the keys you press, in order, to get y = x/4 - 3 onto a Desmos line.
Show the answer
Answer: y = x, then /, then 4, then the right-arrow key, then - 3.
Without the right arrow the - 3 lands in the denominator, so you graph x over (4 - 3), which is just y = x. It looks perfectly tidy on screen, and that is what makes this mistake expensive: nothing tells you it is wrong.
3
A pool holds 200 litres and drains at 4 litres per minute. A second pool holds 80 litres and fills at 6 litres per minute. You graph y = -4x + 200 and y = 6x + 80, and the crossing is at (12, 152). The question asks when the two pools hold the same amount. Which number do you give?
Show the answer
Answer: 12
You set x as minutes, so "when" is the x-coordinate: 12 minutes. Had the question asked how much each pool holds, the answer would have been 152 litres. Same graph, same click, different coordinate. The wording decides, not the picture.
4
In the equation (k/5)x + 2 = 4x - 7, k is a constant and the equation has no solution. What is the value of k?
Show the answer
Answer: 20
No solution means the two sides are parallel lines, so their slopes are equal: k/5 = 4, which gives k = 20. Answering 4 means you stopped at the slope. If you want the arithmetic done for you, type 5*4 on a bare Desmos line.

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