Algebra

Quick Answer: Algebra questions require you to simplify expressions, solve equations and inequalities, work with linear equations, and translate word problems. The key skill is isolating variables by performing inverse operations. A systematic approach—translating words to equations, then solving step-by-step—will help you succeed.

Algebraic Expressions

Terms and Like Terms

Combining Like Terms

Only add/subtract terms with the same variable and exponent.

Example: 5x + 3y - 2x + 7y

• Combine x terms: 5x - 2x = 3x
• Combine y terms: 3y + 7y = 10y
• Answer: 3x + 10y

Distributive Property

$$a(b + c) = ab + ac$$

Example: 3(2x - 5)

• Distribute: 3 × 2x - 3 × 5 = 6x - 15

Worked Example: Simplifying Expressions

Problem: Simplify: 4(2x + 3) - 2(x - 5)

Solution:

1. Distribute: 8x + 12 - 2x + 10
2. Combine like terms: (8x - 2x) + (12 + 10)
3. Answer: 6x + 22

Solving Linear Equations

Golden Rule

Whatever you do to one side of the equation, do to the other side.

Steps to Solve

1. Simplify each side (distribute, combine like terms)
2. Move variable terms to one side
3. Move constant terms to the other side
4. Divide to isolate the variable

Worked Example: One-Step Equation

Problem: Solve: x + 7 = 15

Solution:

• Subtract 7 from both sides: x = 15 - 7 = 8

Worked Example: Two-Step Equation

Problem: Solve: 3x - 4 = 11

Solution:

1. Add 4 to both sides: 3x = 15
2. Divide both sides by 3: x = 5

Worked Example: Multi-Step Equation

Problem: Solve: 2(x + 3) = 5x - 9

Solution:

1. Distribute: 2x + 6 = 5x - 9
2. Subtract 2x from both sides: 6 = 3x - 9
3. Add 9 to both sides: 15 = 3x
4. Divide by 3: x = 5

Check: 2(5 + 3) = 2(8) = 16; 5(5) - 9 = 25 - 9 = 16 ✓

Test Tip: Always check your answer by substituting back into the original equation. This catches arithmetic errors and confirms you solved correctly.

Inequalities

Inequality Symbols

Solving Inequalities

Solve like equations, BUT: Flip the inequality sign when multiplying or dividing by a negative number.

Worked Example: Basic Inequality

Problem: Solve: 2x + 5 > 11

Solution:

1. Subtract 5: 2x > 6
2. Divide by 2: x > 3

Worked Example: Flip the Sign

Problem: Solve: -3x + 6 ≤ 15

Solution:

1. Subtract 6: -3x ≤ 9
2. Divide by -3 (FLIP the sign): x ≥ -3

Linear Equations in Two Variables

Slope-Intercept Form

$$y = mx + b$$

Where:

• m = slope (rate of change)
• b = y-intercept (where line crosses y-axis)

Understanding Slope

$$\text{slope} = m = \frac{\text{rise}}{\text{run}} = \frac{y_2 - y_1}{x_2 - x_1}$$

Worked Example: Finding Slope

Problem: Find the slope of the line passing through (2, 3) and (6, 11).

Solution:

$$m = \frac{11 - 3}{6 - 2} = \frac{8}{4} = \textbf{2}$$

Worked Example: Writing an Equation

Problem: Write the equation of a line with slope 3 that passes through (0, -2).

Solution:

• Given: m = 3, passes through (0, -2)
• Since x = 0, the point is the y-intercept: b = -2
• Equation: y = 3x - 2

Word Problems

Translation Guide

Worked Example: Age Problem

Problem: A teacher is 4 times as old as a student. In 10 years, the teacher will be 2.5 times as old as the student. How old is the student now?

Solution:

1. Define variable: Let s = student's current age

2. Set up expressions:

   • Teacher's current age: 4s
   • Student's age in 10 years: s + 10
   • Teacher's age in 10 years: 4s + 10

3. Write equation: 4s + 10 = 2.5(s + 10)

4. Solve:

   • 4s + 10 = 2.5s + 25
   • 4s - 2.5s = 25 - 10
   • 1.5s = 15
   • s = 10 years old

Check: Student is 10, teacher is 40. In 10 years: student is 20, teacher is 50. Is 50 = 2.5 × 20? Yes! ✓

Worked Example: Mixture Problem

Problem: A teacher mixes a 20% acid solution with a 50% acid solution to get 6 liters of a 30% acid solution. How many liters of the 20% solution were used?

Solution:

1. Define variable: Let x = liters of 20% solution
2. Express other quantity: 6 - x = liters of 50% solution
3. Set up equation (acid amounts):
   • 0.20x + 0.50(6 - x) = 0.30(6)
4. Solve:
   • 0.20x + 3 - 0.50x = 1.8
   • -0.30x = -1.2
   • x = 4 liters of 20% solution

Calculator Tip: For word problems, the calculator helps with decimal arithmetic. But set up your equation first on scratch paper, then use the calculator for computation.