Algebra — Summary

1 Variables & Expressions

Key Vocabulary

Term	Meaning	Example
Variable	A letter representing an unknown number	x, y, n
Term	A single number, variable, or product of both	5x, −3, 7xy
Coefficient	The number multiplying a variable	In 5x, the coefficient is 5
Like terms	Terms with the same variable(s) and exponents	3x and −7x are like terms
Expression	A combination of terms (no = sign)	3x + 2y − 5
Equation	Two expressions linked by = sign	3x + 5 = 14

The Distributive Property

Distributive law

a(b + c) = ab + ac

✏️

Expand: 3(2x − 5)
= 3 × 2x − 3 × 5 = 6x − 15

Expand: −2(x + 4)
= −2 × x + (−2) × 4 = −2x − 8

2 Simplifying Expressions

Combining Like Terms

Only like terms can be added or subtracted. Combine their coefficients.

✏️

Simplify: 5x + 3y − 2x + 7 − y
x terms: 5x − 2x = 3x
y terms: 3y − y = 2y
constants: 7
Result: 3x + 2y + 7

Expanding then Simplifying

✏️

Simplify: 2(3x + 1) − 4(x − 2)
Step 1 — Expand: 6x + 2 − 4x + 8
Step 2 — Combine like terms: (6x − 4x) + (2 + 8) = 2x + 10

⚠️When distributing a negative: −4(x − 2) = −4x + 8, not −4x − 8. Multiplying two negatives gives a positive.

3 First-Degree Equations

🔑Goal: isolate the variable by performing the same operation on both sides of the equation.

Solving Steps

Expand any brackets
Collect all variable terms on one side
Collect all constant terms on the other side
Divide by the coefficient
Check your answer by substituting back

✏️

Solve: 3(x − 2) = 2x + 7
Step 1: 3x − 6 = 2x + 7
Step 2: 3x − 2x = 7 + 6
Step 3: x = 13
Check: 3(13 − 2) = 3 × 11 = 33 and 2(13) + 7 = 33 ✓

Equations with Fractions

Multiply every term by the LCD to clear all denominators first.

✏️

Solve: x/3 + 1 = x/2 − 2
LCD = 6. Multiply everything by 6:
2x + 6 = 3x − 12
6 + 12 = 3x − 2x
x = 18

4 First-Degree Inequalities

Inequalities work just like equations, with one crucial difference.

⚠️When you multiply or divide both sides by a negative number, flip the inequality sign.
−2x < 8 → x > −4 (sign flipped because we divided by −2)

Inequality Symbols

Symbol	Meaning	Graph on number line
<	strictly less than	Open circle, arrow left
>	strictly greater than	Open circle, arrow right
≤	less than or equal to	Closed circle, arrow left
≥	greater than or equal to	Closed circle, arrow right

✏️

Solve: 5 − 3x ≥ 14
−3x ≥ 14 − 5
−3x ≥ 9
x ≤ −3 (flip sign — divided by −3)
Solution: all x ≤ −3

5 Word Problem Strategy

💡

5-step method:
1. Read carefully — identify what is unknown
2. Define your variable (let x = …)
3. Write an equation using the given information
4. Solve the equation
5. Answer the question in a full sentence with units

✏️

A number increased by 8 equals three times the number minus 4. Find the number.
Let x = the number
x + 8 = 3x − 4
8 + 4 = 3x − x
12 = 2x
x = 6
The number is 6.

6 Common Mistakes to Avoid

Mistake	What to do instead
Not distributing to every term	3(x + 2) = 3x + 6, not 3x + 2. Multiply every term inside the brackets.
Dropping the sign when moving terms	3x − 5 = 7 → 3x = 7 + 5 = 12. The sign changes when crossing the = sign.
Forgetting to flip inequality	Always flip the inequality symbol when multiplying or dividing by a negative.
Combining unlike terms	3x + 2y ≠ 5xy. Only like terms (same variable, same exponent) can be combined.
Not checking the answer	Always substitute back into the original equation to verify.
Negative × negative error	−3(x − 4) = −3x + 12, not −3x − 12. Two negatives make a positive.