Solve linear equations

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Lesson progressPractice problems 0/4

Difficulty: Beginner
Estimated time: 30 minutes
Domains: Algebra
Techniques: Equivalent-stepsRepeated-expressionMethod-choiceRequested-expression

What you’ll learn

Solve a linear equation that has one solution, keeping both sides equal at every step.
Handle variables on both sides, parentheses, fractions and decimals.
Choose between a few quick steps by hand, typing the equation straight into Desmos, and graphing both sides.
Use the equation’s structure to find exactly the value the question asks for.

Prerequisites

You’re ready. No earlier Aniko lesson is required.

Why this matters on the SAT

Find x, then answer the question

Linear equations show up all over SAT Math. Some take two quick steps. Others come wrapped in a word problem, packed with fractions or decimals, or built around one expression that appears on both sides. The job is the same every time. Keep both sides equal, watch for a shortcut in the structure, and give the value the question asks for, which isn’t always $x$ .

Here’s one where Desmos does the solving and you do the thinking.

SAT example

The number $x$ satisfies

\frac{2x+5}{3}-\frac{x-1}{4}=9.

What is the value of $x-4$ ?

A
$4$
B
$9$
C
$13$
D
$17$

Solution to the example

Clearing those fractions by hand takes several careful steps. Desmos can skip them, so type the whole equation in exactly as it’s written:

(2x+5)/3-(x-1)/4=9

Desmos draws a vertical line. Every point on it has the same $x$ -value, and that value is the solution. Click the line to read $x=17$ . If the line is hard to click, type the left side and the right side as two separate $y=$ lines and click the point where they cross.

Now for the trap. You found $x=17$ , but the question asks for $x-4$ :

17-4=13.

The answer is C. Choice D is $17$ , the value of $x$ , and it’s there for anyone who stops one step early. Desmos did the solving, but reading the last line of the question was still your job.

Calculator loads as you approach

This is the equation from the example. Click its vertical line to read

x=17

, then edit the equation to try the same steps yourself.

Keep both sides equal

An equation says that the left side and the right side have the same value. Every step you take has to keep that true. If you change only one side, you’ve written a different equation.

These moves keep both sides equal:

Add or subtract the same amount on both sides.
Multiply or divide both sides by the same amount, as long as it isn’t zero.
Distribute, which means multiplying the number outside the parentheses by every term inside.
Combine like terms, which have the same variable part, such as $4x$ and $2x$ . Plain numbers combine with each other too.

Here’s how those moves solve

4(x-3)+7=2x+11.

\begin{aligned} 4x-12+7&=2x+11\\[1.4em] 4x-5&=2x+11\\[1.4em] 2x-5&=11\\[1.4em] 2x&=16\\[1.4em] x&=8. \end{aligned}

The first line distributes the $4$ to both terms in the parentheses. The second combines $-12$ and $7$ . After that, you subtract $2x$ from both sides, add $5$ to both sides, and divide both sides by $2$ . Each new line is equivalent to the one before it, which means it has the same solution. To check, put $x=8$ back into the original equation. Both sides come out to $27$ .

Check your understanding:

Solve $5x+4=3x+18$ , doing the same thing to both sides at each step. What is $x$ ? And what goes wrong if you subtract $3x$ from the left side only?

Common mistake:

A common slip is writing $4(x-3)$ as $4x-3$ . The $4$ reached the $x$ but never got to the $3$ . The number outside the parentheses multiplies every term inside, so $4(x-3)=4x-12$ . To check, try $x=2$ : $4(2-3)$ and $4x-12$ both give $-4$ , but $4x-3$ gives $5$ .

Pick the shorter method

Before you start solving, take a few seconds to look at the equation. What it looks like tells you where to solve it.

If the equation is full of fractions or decimals, type the whole thing into Desmos first, exactly as it’s written.
If one or two quick steps will finish it, do it by hand. For $3x+5=20$ , you subtract $5$ and divide by $3$ .
If an expression like $5-2x$ shows up unchanged on both sides, or the question asks for a multiple of one whole side, use that structure by hand.
If the equation is long, with lots of minus signs or parentheses inside parentheses, and no shortcut jumps out, typing it into Desmos also lowers the risk of an arithmetic slip.

Here’s one full of fractions:

\frac78(x-4)+3=\frac38(x+4)+12.

Type the whole equation in, using a lowercase $x$ :

(7/8)(x-4)+3=(3/8)(x+4)+12

Desmos draws a vertical line at $x=28$ . When the line is clear, click it and read its $x$ -value.

When the line is hard to read, graph both sides. Sometimes the vertical line is hard to click or read, or you want to see how the two sides compare. Then graph each side of the equation as its own line. Here $y$ is only a label for the value one side gives at a chosen $x$ . Where the two lines cross, the same $x$ gives both sides the same value, and that’s exactly what it means for $x$ to solve the equation.

To graph both sides:

Type the left side as y=(7/8)(x-4)+3.
Type the right side as y=(3/8)(x+4)+12.
Click the point where the two lines cross.
Read its first coordinate, the $x$ -coordinate.

The lines cross at $(28,24)$ . The first coordinate, $28$ , is the input both sides share, so the solution is $x=28$ . The second coordinate, $24$ , is the value both sides reach there. It isn’t what this equation asks you to find, so don’t let it slip into your answer.

Both Desmos methods give you a number read off a graph. If Desmos shows a decimal that never ends and the question wants an exact fraction, work out the exact value by hand, or use the answer choices to find the exact value and check it.

Try it yourself:

Change the last $12$ on the right side to $10$ . Will the solution move left or right? Make your guess, then check it in the calculator.

Check your understanding:

Which equation would you start by hand: $3x+5=20$ or $\frac{11}{12}(x-7)-\frac25=\frac13(x+8)$ ? Why?

Related: For more calculator practice, try Solve one-variable equations in the Desmos course. It’s optional, and you don’t need it for anything here.

Calculator loads as you approach

Click the vertical line to read

x=28

. If it’s hard to read, edit the equation into two

y=

lines, one for each side.

Example: Solve for the repeated expression

Worked example

7(5-2x)-4=6(5-2x)+9,

what is the value of $5-2x$ ?

A
$-13$
B
$4$
C
$9$
D
$13$

Step 1

Spot what repeats

Look at $5-2x$ . It appears twice, unchanged, and it’s exactly what the question asks for. So treat the whole expression as one unknown and give it a short name:

u=5-2x.

The letter $u$ is only a nickname for $5-2x$ . It isn’t a new number you have to find separately.

Step 2

Solve the smaller equation

Swap each $5-2x$ for $u$ . Then solve the way you always do, keeping both sides equal:

\begin{aligned} 7u-4&=6u+9\\[1.4em] u-4&=9\\[1.4em] u&=13. \end{aligned}

Subtracting $6u$ from both sides gives the second line. Adding $4$ to both sides gives $u=13$ .

Step 3

Answer the question

Since $u=5-2x$ ,

5-2x=\boxed{13}.

The answer is D. You never needed $x$ . The question asks for $5-2x$ , so finding $x$ first would only add steps before you got back to the same $13$ .

Common mistake:

If your work fills up with separate $x$ -terms, you expanded $5-2x$ before noticing that it repeats. It’s an easy habit to fall into, since expanding is often the first move. Here it only makes the algebra longer. Keep $5-2x$ together, call it $u$ , and solve the short equation. To check $u=13$ , notice that both sides of the original equation then come out to $87$ .

When the question asks for a multiple of one side

Before you solve for $x$ , compare what the question asks for with each whole side of the equation. If it’s a multiple of one side, scale the whole equation, which means multiplying both whole sides by the same number.

Say $2x+3=11$ and the question asks for $6x+9$ . That’s $3$ times the left side, so

6x+9=3(2x+3)=3(11)=33.

You get the answer in one step. Finding $x$ first would only add work.

Practice problems

Your turn. Size up each equation before you pick a method, and reread the last line of the question before you answer.

Equation with parentheses

Practice problem

Which value of $x$ satisfies the equation below?

3(2x-5)+4=4x+9

Calculator loads as you approach

Type the equation in as it’s written. There’s no need to distribute first.

Fraction equation

Practice problem

The number $x$ satisfies

\frac56(x+3)-\frac7{12}(x-6)=13.

What is the value of $\frac{x}{7}$ ?

Your answer

Calculator loads as you approach

Type the equation on one line and read the solution. Then finish the question.

Related expression

Practice problem

The number $x$ satisfies

5-3x=12.

What is the value of $15-9x$ ?

Your answer

Calculator loads as you approach

The calculator is here if you want to check your work. Make sure your answer is the value the question asks for.

Nested expression

Practice problem

5-2\bigl[3-4(2x+1)\bigr] = 18-3\bigl[3-4(2x+1)\bigr],

what is the value of $3-4(2x+1)$ ?

Your answer

Calculator loads as you approach

This one is faster by hand if you keep the repeated bracket in one piece.

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Keep both sides equal: do the same thing to both sides, distribute to every term, and combine only like terms.
When a shortcut is easy to see, use it by hand. That means one or two quick steps, a multiple of one whole side, or a repeated expression.
Type an equation full of fractions or decimals straight into Desmos. A long one with lots of signs or nested parentheses and no clear shortcut usually goes there too, as long as the vertical line is easy to read.
Graph both sides when the vertical line is hard to read or you want to see how the two sides compare, and read the $x$ -coordinate where they cross.
Find $x$ , then answer the question. Reread the last line before you submit, since Desmos only shows you the solution.

Next lesson

Model and interpret one-variable equations

Turn a context into a linear equation and explain what its terms and solution mean.

Start next lesson

Practice

Practice this lesson

1,298 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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