Solve one-variable equations

Your feedback matters:

Lesson progressPractice problems 0/4

Difficulty: Beginner
Estimated time: 35 minutes
Domains: Algebra Advanced Math
Techniques: IntersectionsSolve

What you’ll learn

Recognize when a question gives one equation with one unknown.
Graph both sides and read every real solution from their intersections.
Use direct equation entry as a shortcut when it is clear and reliable.
Replace another variable letter with x without changing the equation.
Apply conditions such as positive, least, or greatest.
Represent a two-category total with one unknown before graphing.
Calculate the requested expression after finding the solution.
Choose a shorter hand method when Desmos adds unnecessary work.

Prerequisites

Why this matters on the SAT

Solve long equations without rearranging

SAT equations can hide a simple answer behind fractions, parentheses, and several algebra steps. When an equation contains one unknown, Desmos can often solve it without rearranging the algebra first.

SAT example

Which choice is the solution to the equation?

\frac{5}{6}(x-3)+4=\frac{1}{3}(x+6)+11

A
$17$
B
$20$
C
$23$
D
$26$

Fast Desmos solution

Graph y=(5/6)(x-3)+4 and y=(1/3)(x+6)+11. The graphs intersect where $x=23$ . That $x$ -coordinate solves the original equation, so the answer is C.

The SAT may ask for the positive solution, the greater solution, or an expression built from the solution. Solving the equation is only the first part. You must still answer the specific quantity requested.

Calculator loads as you approach

The x-coordinate of the intersection solves the original equation.

When should you solve an equation in Desmos?

Desmos is strongest when one unknown is buried inside algebra that would be slow to rearrange by hand.

Begin with Desmos when…

the equation contains one unknown;
fractions, parentheses, powers, or absolute values make algebra slow;
the equation may have more than one real solution;
the prompt asks for a positive, negative, least, or greatest solution; or
the prompt asks for a value calculated from the solution.

Use algebra when…

the equation takes only one or two quick steps. For $\frac{x}{4}+3=8$ , subtract $3$ and multiply by $4$ to get $x=20$ .

Look for wording such as “positive solution,” “one possible solution,” or “if $x$ is the solution, find $x-4$ .”

Check your understanding:

Which equation is a stronger reason to begin with Desmos: $x+8=13$ or $\lvert 3x-4\rvert=20$ ? Explain the difference.

1. Graph the two sides

For an equation

L(x)=R(x),

a solution is an input where the left side and right side have the same value. Graph y=L(x) and y=R(x). At each intersection, both sides have the same output. The intersection’s $x$ -coordinate solves the original equation.

For

5(x-4)+7=3(x+2)+5,

graph y=5(x-4)+7 and y=3(x+2)+5. The graphs intersect at $(12,47)$ , so the solution is $x=12$ .

Try it yourself:

Change the final $+5$ in the right-side graph to $+9$ . Predict whether the intersection will move left or right, then select the new point to check.

Check your understanding:

The graphs above intersect at $(12,47)$ . Which coordinate solves the original equation, and what does the other coordinate mean?

The original problem has one unknown, $x$ . Desmos uses $y$ only as a temporary comparison value. Lesson 4 covers systems in which both $x$ and $y$ are original unknowns.

Replace other variable letters with x

Desmos uses lowercase $x$ as its horizontal graphing variable. If the equation uses another letter, replace every occurrence of that unknown with $x$ .

For $3z+8=29$ , graph y=3x+8 and y=29. Their intersection has $x$ -coordinate $7$ , which means the original variable is $z=7$ .

Common mistake:

Replacing only some occurrences of the original variable. If the equation contains $t$ three times, all three must become $x$ .

Calculator loads as you approach

The shared y-value shows that the two sides are equal; the x-value is the solution.

2. Keep every solution

The same two-graph method reveals when an equation has more than one real solution. Pan or zoom to inspect the relevant graph before deciding that you have found every solution. Then apply words such as positive, least, or greatest.

For

\lvert 2x+1\rvert=11,

graph y=|2x+1| and y=11. The graphs intersect at

(-6,11)\qquad\text{and}\qquad(5,11).

The solutions are their $x$ -coordinates:

x=-6\qquad\text{and}\qquad x=5.

Try it yourself:

Before selecting either intersection, predict how many solutions the V-shaped graph and horizontal line will produce. Then change $11$ to $5$ and predict how the solutions will move.

Use the two-graph method when:

an equation may have multiple solutions;
the question asks you to compare the solutions;
the shape of each side helps explain the result; or
you want a clear visual check that the two sides are equal.

Common mistake:

Reporting the shared $y$ -coordinate. Here, $11$ proves that the two sides are equal, but the equation’s solutions are the intersection $x$ -coordinates.

Check your understanding:

For $\lvert 2x+1\rvert=11$ , what is the least solution, and which part of an intersection gives that answer?

Graph labels are numerical. If Desmos shows a nonterminating decimal but the question requires an exact fraction or radical, do not guess the exact form from the display. Use fraction conversion when it gives a clear result, verify an answer choice, or use exact algebra. Lesson 27 develops the full exactness and output-verification workflow.

Questions about whether an equation has no solutions, one solution, or infinitely many solutions require extra care. Lesson 8 develops the full solution-count workflow.

Direct entry can be a shortcut

You can also enter the complete equation on one expression line. Entering |2x+1|=11 graphs vertical solution lines at $x=-6$ and $x=5$ . This can save time when the lines are easy to select. Graph both sides when you want the clearest view of the equation’s structure or all of its solutions.

Keep the original equation intact. Do not cancel variable factors or square both sides merely to make the graph easier to enter. Those changes can remove restrictions or create invalid candidates. Lesson 15 develops restrictions and extraneous-solution checks.

Calculator loads as you approach

The x-coordinate of each intersection is a solution to the original equation.

3. Solve first, then answer the question

Worked example

The greater solution to the equation

(x-3)^2=5x-9

is $x$ .

What is the value of $2x+1$ ?

Step 1

Identify both jobs

First find every real solution. Then choose the greater solution and evaluate $2x+1$ . The solution itself is not the final answer.

Step 2

Graph both sides

Enter y=(x-3)^2 and y=5x-9 on separate expression lines. The graphs intersect at

(2,1)\qquad\text{and}\qquad(9,36).

Their $x$ -coordinates give the solutions $x=2$ and $x=9$ .

Calculator loads as you approach

Each intersection x-coordinate satisfies the original equation.

Step 3

Apply the condition

The two solutions are $2$ and $9$ . The greater solution is

x=9.

Step 4

Evaluate the requested expression

Substitute $9$ into the expression the question asks for:

2x+1=2(9)+1=\boxed{19}.

Common mistake:

Submitting $9$ . That is the correct solution to the equation, but the question asks for $2x+1$ .

4. Turn a context into one equation

Some SAT questions describe a situation instead of giving the final equation. Choose one unknown, express every changing quantity with that unknown, and then enter the resulting equation.

If a venue sells $n$ tickets in total and $x$ are adult tickets, then the number of student tickets is

n-x.

If adult tickets cost $a$ dollars and student tickets cost $s$ dollars, the total revenue equation is

ax+s(n-x)=\text{total revenue}.

Once the equation represents every stated condition, graph both sides as usual.

Check your understanding:

A center packs $62$ boxes in total. If $x$ is the number of large boxes, how should the number of small boxes be written?

Common mistake:

Using $x$ for both categories. If $x$ is the number of adult tickets, it cannot also be the number of student tickets.

Finish the solution

The equation is already graphed with $x$ in place of $z$ . Select the intersection, then submit the value the question asks for.

Solve, then divide

Finish the solution

The number $z$ is the solution to

\frac{3z+1}{5}-\frac{z-2}{2}=3.

What is the value of $\frac{z}{3}$ ?

First steps

Replace every $z$ with $x$ .
Graph the left side and right side on separate expression lines.
Select their intersection.

Finish it

Your answer

Calculator loads as you approach

Read the intersection x-coordinate, then evaluate the requested expression.

Practice problems

SAT practice problems

Graph both sides unless direct entry is clearly faster. Keep every solution visible, apply the prompt’s condition, and submit only the requested value.

Solve a linear equation

Practice problem

Which choice is the solution to the equation?

4(x+3)-5=2(x-1)+17

Calculator loads as you approach

Graph the two sides and select their intersection.

Choose the positive solution

Practice problem

What is the positive solution to the equation?

\lvert 3x-2\rvert=16

Calculator loads as you approach

Find both intersections before applying the positive-solution condition.

Build a one-variable equation

Practice problem

A theater sold $84$ tickets for a performance. Adult tickets cost $18 each, and student tickets cost $11 each. The theater collected $1,204 from these tickets.

How many adult tickets were sold?

Calculator loads as you approach

Represent the two ticket categories with one unknown, then graph both sides.

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Graph the left side and right side on separate expression lines.
Each intersection $x$ -coordinate solves the original equation.
Direct equation entry can be faster when its vertical solution lines are easy to read.
Replace another unknown letter with $x$ consistently.
Inspect the relevant graph broadly, then keep every real solution before applying words such as positive, least, or greatest.
After solving, calculate the expression or contextual quantity the question requests.
Use a short hand solution when Desmos would add work.

Next lesson

Solve systems at intersections

Graph two equations with two original unknowns and read their shared solution point.

Start next lesson

Free Practice

What you’ll learn

Prerequisites

Solve long equations without rearranging

Fast Desmos solution

When should you solve an equation in Desmos?

Begin with Desmos when…

Use algebra when…

1. Graph the two sides

Replace other variable letters with x

2. Keep every solution

Direct entry can be a shortcut

3. Solve first, then answer the question

Identify both jobs

Graph both sides

Apply the condition

Evaluate the requested expression

4. Turn a context into one equation

Finish the solution

Solve, then divide

First steps

Finish it

Practice problems

SAT practice problems

Solve a linear equation

Choose the positive solution

Build a one-variable equation

4 practice examples left

Quick recap

Solve systems at intersections

Free question bank practice problems