Tier 1: Expressions & Linear Equations

Level 1.1 – Like Terms

Combine like terms to simplify algebraic expressions.

🧠 What Are Like Terms?

Like terms are terms that have the same variables raised to the same power. For example:

3x and 2x are like terms
5xy and -2xy are like terms
3x and 3x² are not like terms

✏️ Example Problem

Simplify the expression: 3x + 5x - 2x

Solution: Combine all the x terms: 3x + 5x - 2x = 6x

✅ Try It Yourself

What is 4x + 2y - x + 3y simplified?

Answer: Group x terms and y terms → 4x - x = 3x, 2y + 3y = 5y
Final Answer: 3x + 5y

Level 1.2 – Distributive Property

Apply the distributive law: a(b + c) = ab + ac.

🧠 What Is the Distributive Property?

The distributive property allows you to multiply a value across terms inside parentheses:

Rule: a(b + c) = ab + ac

✏️ Examples

3(x + 4) = 3x + 12
-2(x - 5) = -2x + 10
4(2x + 3) = 8x + 12
-5(3a - 2b) = -15a + 10b

✅ Try It Yourself

Simplify the following:

2(x + 6) → Answer: 2x + 12
-3(y - 4) → Answer: -3y + 12
6(2m + 5) → Answer: 12m + 30

Level 1.3 – One-Step Equations

Solve equations in one operation. Example: x - 4 = 6 → x = 10

🧠 What Are One-Step Equations?

A one-step equation can be solved with just one inverse operation like addition, subtraction, multiplication, or division.

➕ Solving by Addition

Example: x - 4 = 6
Add 4 to both sides → x = 10

➖ Solving by Subtraction

Example: x + 7 = 12
Subtract 7 from both sides → x = 5

✖️ Solving by Multiplication

Example: x / 3 = 9
Multiply both sides by 3 → x = 27

➗ Solving by Division

Example: 5x = 20
Divide both sides by 5 → x = 4

✅ Try These

x + 8 = 15 → Answer: x = 7
x - 6 = 11 → Answer: x = 17
x / 4 = 5 → Answer: x = 20
7x = 35 → Answer: x = 5

Level 1.4 – Two-Step Equations

Solve using two operations. Example: 2x + 3 = 11 → x = 4

🧠 What Are Two-Step Equations?

A two-step equation requires you to perform two inverse operations to isolate the variable.

General form: ax + b = c

🔧 Steps to Solve

Undo addition or subtraction
Then undo multiplication or division

✏️ Example 1

2x + 3 = 11

Step 1: Subtract 3 → 2x = 8
Step 2: Divide by 2 → x = 4

✏️ Example 2

-3x - 5 = 10

Step 1: Add 5 → -3x = 15
Step 2: Divide by -3 → x = -5

✅ Try These

4x - 7 = 9 → Answer: x = 4
-5x + 2 = -13 → Answer: x = 3
6x + 12 = 0 → Answer: x = -2

Level 1.5 – Variables on Both Sides

Move variables to one side and solve. Example: 3x + 2 = x + 6 → x = 2

🧠 What to Do When Variables Are on Both Sides?

Your goal is to collect all variable terms on one side of the equation and constants on the other.

🔧 Strategy

Subtract or add the smaller variable term to move variables to one side
Then solve the resulting one- or two-step equation

✏️ Example 1

3x + 2 = x + 6

Step 1: Subtract x from both sides → 2x + 2 = 6
Step 2: Subtract 2 → 2x = 4
Step 3: Divide by 2 → x = 2

✏️ Example 2

5x - 7 = 3x + 1

Step 1: Subtract 3x → 2x - 7 = 1
Step 2: Add 7 → 2x = 8
Step 3: Divide → x = 4

✅ Try These

4x + 3 = 2x + 9 → Answer: x = 3
6x - 5 = 3x + 1 → Answer: x = 2
7x + 4 = 5x + 12 → Answer: x = 4

Level 1.6 – Fractions in Equations

Clear fractions and solve. Example: (1/2)x = 3 → x = 6

🧠 Why Are Fractions Tricky?

Fractions can make equations feel harder — but we can eliminate them by multiplying both sides by a common denominator.

🔧 Strategy

Find the denominator(s)
Multiply both sides of the equation by that denominator
Solve the resulting simpler equation

✏️ Example 1

(1/2)x = 3

Multiply both sides by 2 → x = 6

✏️ Example 2

x/4 + 2 = 5

Step 1: Subtract 2 → x/4 = 3
Step 2: Multiply both sides by 4 → x = 12

✏️ Example 3

(3/5)x - 1 = 2

Step 1: Add 1 → (3/5)x = 3
Step 2: Multiply both sides by 5 → 3x = 15
Step 3: Divide by 3 → x = 5

✅ Try These

x/3 = 7 → Answer: x = 21
(2/3)x = 10 → Answer: x = 15
x/6 - 1 = 2 → Answer: x = 18