Algebra

MK Subtest: About 40% of Mathematics Knowledge questions involve algebra. You have 20 minutes for 16 questions—work efficiently!

Algebraic Expressions

Key Terms

Combining Like Terms

Only add or subtract terms with the same variable(s) and exponent(s).

Example: Simplify $4x^2 + 3x - 2x^2 + 5x - 7$

Solution:

• Combine $x^2$ terms: $4x^2 - 2x^2 = 2x^2$
• Combine $x$ terms: $3x + 5x = 8x$
• Result: $2x^2 + 8x - 7$

Solving Linear Equations

The Goal: Isolate the Variable

Steps:

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 by the coefficient

Worked Example: One-Step Equation

Problem: $x + 7 = 15$

Solution: Subtract 7 from both sides:

$$x = 15 - 7 = 8$$

Worked Example: Two-Step Equation

Problem: $3x - 5 = 16$

Solution: Step 1: Add 5 to both sides: $3x = 21$ Step 2: Divide by 3: $x = 7$

Worked Example: Multi-Step Equation

Problem: $2(x + 3) - 4 = 3x + 1$

Solution: Step 1: Distribute: $2x + 6 - 4 = 3x + 1$ Step 2: Simplify: $2x + 2 = 3x + 1$ Step 3: Subtract $2x$: $2 = x + 1$ Step 4: Subtract 1: $x = 1$

Check: $2(1 + 3) - 4 = 3(1) + 1$ → $8 - 4 = 4$ ✓

Solving Inequalities

Inequality Symbols

Important Rule

When you multiply or divide both sides by a negative number, FLIP the inequality sign!

Worked Example: Solving an Inequality

Problem: $-3x + 7 > 19$

Solution: Step 1: Subtract 7: $-3x > 12$ Step 2: Divide by -3 (FLIP the sign!): $x < -4$

Exponent Rules

Worked Example: Exponents

Problem: Simplify $\frac{x^5 \times x^3}{x^2}$

Solution: Step 1: Product rule in numerator: $\frac{x^8}{x^2}$ Step 2: Quotient rule: $x^{8-2} = x^6$

Polynomials

Multiplying Binomials (FOIL)

FOIL: First, Outer, Inner, Last

Example: $(x + 3)(x + 5)$

• First: $x \times x = x^2$
• Outer: $x \times 5 = 5x$
• Inner: $3 \times x = 3x$
• Last: $3 \times 5 = 15$

Result: $x^2 + 5x + 3x + 15 = x^2 + 8x + 15$

Common Factoring Patterns

Worked Example: Factoring

Problem: Factor $x^2 - 9$

Solution: This is a difference of squares: $a^2 - b^2 = (a+b)(a-b)$

$$x^2 - 9 = x^2 - 3^2 = (x+3)(x-3)$$

Systems of Equations

Substitution Method

1. Solve one equation for one variable
2. Substitute into the other equation
3. Solve and back-substitute

Worked Example: Substitution

Problem: Solve: $y = 2x + 1$ and $3x + y = 11$

Solution: Step 1: Substitute $y = 2x + 1$ into second equation:

$$3x + (2x + 1) = 11$$

Step 2: Solve: $5x + 1 = 11$ → $5x = 10$ → $x = 2$

Step 3: Find y: $y = 2(2) + 1 = 5$

Answer: $(x, y) = (2, 5)$