Before you read

Answer first — the lesson below explains it.

Mia works two part-time jobs, earning $$2$ per hour at one and $$3$ per hour at the other. After deducting $$4$ for transportation, Mia takes home $$16$ . If $x$ is the number of hours worked at each job, the equation $2 x + 3 x - 4 = 16$ applies. What is $x$ ?

A
$4$
B
$\frac{12}{5}$
C
$20$
D
$- 4$

Check your answer

Answer:

Before you read

Answer first — the lesson below explains it.

A subscription box ships $4$ identical packages, each containing $x$ items plus $2$ bonus samples. If the total number of items shipped is $28$ , the equation $4 (x + 2) = 28$ applies. What is $x$ ?

A
$5$
B
$6$
C
$\frac{26}{4}$
D
$7.5$

Check your answer

Answer:

Theory

Combining Like Terms

Before isolating $x$ , you often need to simplify each side of the equation by combining like terms.

Like terms have the same variable raised to the same power:
- $3x$ and $5x$ are like terms $\to$ $3x + 5x = 8x$
- $7$ and $-2$ are like terms $\to$ $7 + (-2) = 5$
- $3x$ and $3x^2$ are NOT like terms

$2(x-4)+3x=17$
$2x-8+3x=17$ distribute
$5x-8=17$ combine like terms
$5x=25$ add 8
$x=5$ divide by 5

Multi-step equation with distribution

Example 1

Solve $4x + 3 + 2x = 21$

Step 1: Combine like terms on the left side:

$4x + 2x + 3 = 21$

$6x + 3 = 21$

Step 2: Subtract $3$ from both sides:

$6x = 18$

Step 3: Divide by $6$ :

$x = 3$

x = 3

4(3) + 3 + 2(3) = 12 + 3 + 6 = 21

✓

Quick check

You just read it. Can you apply it?

Two ride-share services charge differently. Service A charges $$5$ per mile plus a $$3$ base fee. Service B charges $$2$ per mile plus an $$18$ base fee. For how many miles $x$ do both services cost the same? ( $5 x + 3 = 2 x + 18$ )

A
$3$
B
$5$
C
$7$
D
$- 5$

Check your answer

Answer:

Subtract $2 x$ : $3 x + 3 = 18$
Subtract $3$ : $3 x = 15$
Divide by $3$ : $x = 5$
Check: $5 (5) + 3 = 28$ and $2 (5) + 18 = 28$ ✓

Theory

The Distributive Property

When an equation contains parentheses, use the distributive property to expand them first:
$a(b + c) = ab + ac$

Careful with negative signs in front of parentheses:
$-(b + c) = -b - c$
$-2(x - 3) = -2x + 6$

Example 1

Solve $3(x + 4) = 27$

Step 1: Distribute the $3$ :

$3x + 12 = 27$

Step 2: Subtract $12$ :

$3x = 15$

Step 3: Divide by $3$ :

$x = 5$

x = 5

3(5 + 4) = 3(9) = 27

✓

Example 2

Solve $2(3x - 1) + 4 = 16$

Step 1: Distribute the $2$ :

$6x - 2 + 4 = 16$

Step 2: Combine like terms:

$6x + 2 = 16$

Step 3: Subtract $2$ :

$6x = 14$

Step 4: Divide by $6$ :

$x = \frac{14}{6} = \frac{7}{3}$

x = \frac{7}{3}

2(3 \cdot \frac{7}{3} - 1) + 4 = 2(7 - 1) + 4 = 12 + 4 = 16

✓

Quick check

You just read it. Can you apply it?

Aiden calculates a temperature conversion using the formula $- 2 (x - 5) = 14$ , where $x$ is the original reading. What is $x$ ?

A
$- 2$
B
$2$
C
$12$
D
$- 12$

Check your answer

Answer:

Distribute $- 2$ : $- 2 x + 10 = 14$
Subtract $10$ : $- 2 x = 4$
Divide by $- 2$ : $x = - 2$
Check: $- 2 (- 2 - 5) = - 2 (- 7) = 14$ ✓

Theory

Variables on Both Sides

When $x$ appears on both sides of the equation, collect all $x$ -terms on one side and all constants on the other.

Strategy: move the smaller $x$ -term to the side with the larger $x$ -term to avoid negative coefficients (though negatives still work!).

Example 1

Solve $5x + 3 = 2x + 18$

Step 1: Subtract $2x$ from both sides (collect $x$ -terms on the left):

$3x + 3 = 18$

Step 2: Subtract $3$ :

$3x = 15$

Step 3: Divide by $3$ :

$x = 5$

x = 5

5(5) + 3 = 28

and

2(5) + 18 = 28

✓

Example 2

Solve $4(x - 2) = 2(x + 3)$

Step 1: Distribute on both sides:

$4x - 8 = 2x + 6$

Step 2: Subtract $2x$ from both sides:

$2x - 8 = 6$

Step 3: Add $8$ :

$2x = 14$

Step 4: Divide by $2$ :

$x = 7$

x = 7

4(7 - 2) = 4(5) = 20

and

2(7 + 3) = 2(10) = 20

✓

Quick check

You just read it. Can you apply it?

A catering company charges $$3$ per person for $2 x + 1$ guests, then applies a $$5$ discount. The final bill is $$16$ . The equation $3 (2 x + 1) - 5 = 16$ models this. What is $x$ ?

A
$3$
B
$\frac{7}{2}$
C
$\frac{20}{6}$
D
$7$

Check your answer

Answer:

Distribute: $6 x + 3 - 5 = 16$
Combine: $6 x - 2 = 16$
Add $2$ : $6 x = 18$
Divide by $6$ : $x = 3$
Check: $3 (2 (3) + 1) - 5 = 3 (7) - 5 = 21 - 5 = 16$ ✓

Common Mistakes

Common Mistakes to Avoid

Forgetting to distribute to ALL terms inside parentheses

$3(x + 4) = 3x + 4$ $\leftarrow$ WRONG! Should be $3x + 12$

Multiply the outside number by EVERY term inside the parentheses.

Sign errors with negative distribution

$-(x - 5) = -x - 5$ $\leftarrow$ WRONG! Should be $-x + 5$

Distributing a negative flips ALL signs inside the parentheses.

Combining unlike terms

$3x + 5 = 8x$ $\leftarrow$ WRONG! $3x$ and $5$ are NOT like terms

Only combine terms that have the exact same variable part.

Quick check

You just read it. Can you apply it?

Carter saves $$7$ per week minus $$4$ in weekly expenses, while Sophia saves $$3$ per week plus a $$12$ head start. After $x$ weeks they have equal savings: $7 x - 4 = 3 x + 12$ . What is $x$ ?

A
$2$
B
$4$
C
$8$
D
$- 4$

Check your answer

Answer:

Subtract $3 x$ : $4 x - 4 = 12$
Add $4$ : $4 x = 16$
Divide by $4$ : $x = 4$
Check: $7 (4) - 4 = 24$ and $3 (4) + 12 = 24$ ✓

Strategy

SAT Strategy Tip

When you see a multi-step equation on the SAT, don't rush. Take 5 seconds to scan the whole equation before starting. If there are parentheses, distribute first. If there are like terms, combine them. Working neatly prevents sign errors — the #1 source of mistakes on these problems.

After reading

Same question as before. What do you say now?

A
$4$
B
$\frac{12}{5}$
C
$20$
D
$- 4$

Check your answer

Answer:

Combine like terms: $5 x - 4 = 16$
Add $4$ : $5 x = 20$
Divide by $5$ : $x = 4$
Check: $2 (4) + 3 (4) - 4 = 8 + 12 - 4 = 16$ ✓

After reading

Same question as before. What do you say now?

A subscription box ships $4$ identical packages, each containing $x$ items plus $2$ bonus samples. If the total number of items shipped is $28$ , the equation $4 (x + 2) = 28$ applies. What is $x$ ?

A
$5$
B
$6$
C
$\frac{26}{4}$
D
$7.5$

Check your answer

Answer:

Distribute: $4 x + 8 = 28$
Subtract $8$ : $4 x = 20$
Divide by $4$ : $x = 5$
Check: $4 (5 + 2) = 4 (7) = 28$ ✓

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