Before you read

Answer first — the lesson below explains it.

Emma pays a $$9$ registration fee for an online learning platform and then pays an additional amount $x$ for course materials. If the total charge is $$14$ , what is the value of $x$ ?

A
$3$
B
$5$
C
$23$
D
$- 5$

Check your answer

Answer:

Before you read

Answer first — the lesson below explains it.

A tutoring center charges $$5$ per session. If Liam paid $$45$ in total, how many sessions did Liam attend? (Let $x$ = number of sessions, so $5 x = 45$ .)

A
$5$
B
$40$
C
$9$
D
$225$

Check your answer

Answer:

Theory

What Is a Linear Equation?

A linear equation is an equation where the variable (usually $x$ ) appears only to the first power — no $x^2$ , no $\sqrt{x}$ , no $\frac{1}{x}$ .

The simplest form is:
$ax + b = c$
where $a$ , $b$ , and $c$ are constants.

Your goal: find the value of $x$ that makes the equation true.

The golden rule: whatever you do to one side, you must do to the other side. This keeps the equation balanced.

$3x + 7 = 22$
$3x = 15$ subtract 7
$x = 5$ divide by 3

Solving a linear equation step by step

Quick check

You just read it. Can you apply it?

Sophia's phone battery drops by $8%$ from its current level and reaches $- 3%$ below the minimum threshold. If $x$ represents the battery level before the drop, and $x - 8 = - 3$ , what is $x$ ?

A
$- 11$
B
$11$
C
$5$
D
$- 5$

Check your answer

Answer:

Add $8$ to both sides:
$x = - 3 + 8 = 5$
Check: $5 - 8 = - 3$ ✓

Theory

One-Step Equations

Some equations only need one operation to solve.

Addition/Subtraction:
If something is added to $x$ , subtract it from both sides.
If something is subtracted from $x$ , add it to both sides.

Multiplication/Division:
If $x$ is multiplied by a number, divide both sides by that number.
If $x$ is divided by a number, multiply both sides by that number.

Example 1

Solve $x + 7 = 12$

We need to isolate $x$ . Since $7$ is added to $x$ , subtract $7$ from both sides:

$x + 7 - 7 = 12 - 7$

$x = 5$

x = 5

5 + 7 = 12

✓

Example 2

Solve $3x = 18$

Here $x$ is multiplied by $3$ . Divide both sides by $3$ :

$\frac{3x}{3} = \frac{18}{3}$

$x = 6$

x = 6

3(6) = 18

✓

Example 3

Solve $\frac{x}{4} = 9$

Here $x$ is divided by $4$ . Multiply both sides by $4$ :

$4 \cdot \frac{x}{4} = 4 \cdot 9$

$x = 36$

x = 36

\frac{36}{4} = 9

✓

Quick check

You just read it. Can you apply it?

A group of $6$ friends splits the cost of a birthday gift equally. Each person pays $$7$ . If $x$ is the total cost of the gift, and $\frac{x}{6} = 7$ , what is $x$ ?

A
$1$
B
$13$
C
$42$
D
$\frac{7}{6}$

Check your answer

Answer:

Multiply both sides by $6$ :
$x = 6 \times 7 = 42$
Check: $\frac{42}{6} = 7$ ✓

Theory

Two-Step Equations

Many equations need two operations to solve. The strategy is:
1. First, undo addition or subtraction (move the constant)
2. Then, undo multiplication or division (isolate $x$ )

Think of it as peeling off layers — you remove them in reverse order.

Example 1

Solve $2x + 5 = 13$

Step 1: Subtract $5$ from both sides:

$2x + 5 - 5 = 13 - 5$

$2x = 8$

Step 2: Divide both sides by $2$ :

$x = 4$

x = 4

2(4) + 5 = 8 + 5 = 13

✓

Example 2

Solve $7 - 3x = 22$

Step 1: Subtract $7$ from both sides:

$-3x = 15$

Step 2: Divide both sides by $-3$ (watch the sign!):

$x = -5$

x = -5

7 - 3(-5) = 7 + 15 = 22

✓

Quick check

You just read it. Can you apply it?

A phone plan charges $$7$ per month plus $$3$ for each gigabyte of data used. If Olivia's monthly bill is $$22$ , the equation $3 x + 7 = 22$ gives the data usage $x$ in gigabytes. What is $x$ ?

A
$5$
B
$\frac{29}{3}$
C
$15$
D
$- 5$

Check your answer

Answer:

Step 1: Subtract $7$ : $3 x = 15$
Step 2: Divide by $3$ : $x = 5$
Check: $3 (5) + 7 = 22$ ✓

Common Mistakes

Common Mistakes to Avoid

Forgetting to flip the sign when dividing by a negative number

$-2x = 10$ $\to$ many students write $x = 5$ instead of $x = -5$

When you divide by a negative, the sign of the result changes.

Subtracting from only one side

$x + 3 = 10$ $\to$ writing $x = 10$ (forgot to subtract 3 from the right)

Always perform the same operation on BOTH sides.

Doing operations in the wrong order for two-step equations

In $2x + 5 = 13$ , dividing by 2 first gives $x + 5 = 6.5$ — messy!

Always undo addition/subtraction first, then multiplication/division.

Quick check

You just read it. Can you apply it?

A gym charges $$4$ per visit and offers a $$10$ discount coupon. After using the coupon, Mason paid $$18$ . The equation $4 x - 10 = 18$ models this situation, where $x$ is the number of visits. What is $x$ ?

A
$2$
B
$7$
C
$\frac{18}{4}$
D
$- 7$

Check your answer

Answer:

Step 1: Add $10$ : $4 x = 28$
Step 2: Divide by $4$ : $x = 7$
Check: $4 (7) - 10 = 18$ ✓

Strategy

SAT Strategy Tip

Basic linear equations are among the easiest SAT questions and appear early in each module. They are free points — solve them quickly and accurately to save time for harder problems. Always check your answer by plugging it back in.

After reading

Same question as before. What do you say now?

Emma pays a $$9$ registration fee for an online learning platform and then pays an additional amount $x$ for course materials. If the total charge is $$14$ , what is the value of $x$ ?

A
$3$
B
$5$
C
$23$
D
$- 5$

Check your answer

Answer:

Subtract $9$ from both sides:
$x = 14 - 9 = 5$
Check: $5 + 9 = 14$ ✓

After reading

Same question as before. What do you say now?

A tutoring center charges $$5$ per session. If Liam paid $$45$ in total, how many sessions did Liam attend? (Let $x$ = number of sessions, so $5 x = 45$ .)

A
$5$
B
$40$
C
$9$
D
$225$

Check your answer

Answer:

Divide both sides by $5$ :
$x = \frac{45}{5} = 9$
Check: $5 (9) = 45$ ✓

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