Algebra

The Framework and Structure

Algebra provides the framing and structure of GRE math, allowing you to solve complex problems using rules. This section covers approximately 25% of Quantitative Reasoning questions.

  • Algebraic expressions and equations
  • Linear and quadratic equations
  • Inequalities and absolute value
  • Functions and their properties
  • Coordinate geometry and graphing
  • Word problems (work, mixture, rate)

Linear Equations in Two Variables

Two Solution Methods

Systems of linear equations can be solved using substitution or elimination.

Method 1: Substitution

Example:

Solve the system:

4x + 3y = 13

x + 2y = 2

Solution:

  1. From the second equation: x = 2 - 2y
  2. Substitute into the first equation: 4(2 - 2y) + 3y = 13
  3. Simplify: 8 - 8y + 3y = 13
  4. Combine terms: -5y = 5
  5. Solve for y: y = -1
  6. Find x: x = 2 - 2(-1) = 2 + 2 = 4
  7. Answer: x = 4, y = -1

Method 2: Elimination

Multiply equations to make coefficients of one variable opposites, then add equations to eliminate that variable.

Quadratic Equations

Standard Form and Formula

Quadratic equations follow the form ax2 + bx + c = 0 and can be solved using the quadratic formula:

x = (-b ± √(b2 - 4ac)) / 2a

Example:

Solve: x2 + 4x + 4 = 0

Solution:

  • Identify: a = 1, b = 4, c = 4
  • Apply formula: x = (-4 ± √(16 - 16)) / 2
  • Simplify: x = (-4 ± √0) / 2
  • Calculate: x = -4 / 2 = -2
  • Answer: Only one real solution, x = -2

Note: This can also be factored as (x + 2)(x + 2) = 0

The Discriminant (b2 - 4ac):

  • If b2 - 4ac > 0: Two distinct real solutions
  • If b2 - 4ac = 0: One repeated real solution
  • If b2 - 4ac < 0: No real solutions (two complex solutions)

Work Problems

Key Concept: Rates

Work problems involve combining individual production rates to find a total time.

Rate = 1 / Time

Combined rate = Rate1 + Rate2 + ...

Example:

Machine A takes 3 hours to produce a batch. Machine B takes 2 hours. How long do they take working together?

Solution:

  • Machine A rate: 1/3 batch per hour
  • Machine B rate: 1/2 batch per hour
  • Combined rate: 1/3 + 1/2 = 2/6 + 3/6 = 5/6 batch per hour
  • Time to complete 1 batch: 1 ÷ (5/6) = 6/5 hours
  • Answer: 6/5 hours = 1 hour and 12 minutes

Coordinate Geometry

Slope Formula

The slope (m) of a line through two points (x1, y1) and (x2, y2) is:

m = (y2 - y1) / (x2 - x1)

Example:

Find the slope of the line through points Q(-2, -3) and R(4, 1.5):

Solution:

  • m = (1.5 - (-3)) / (4 - (-2))
  • m = (1.5 + 3) / (4 + 2)
  • m = 4.5 / 6
  • Answer: m = 0.75

Other Important Formulas

Distance Formula:

d = √((x2-x1)2 + (y2-y1)2)

Midpoint Formula:

M = ((x1+x2)/2, (y1+y2)/2)

Slope-Intercept Form:

y = mx + b

where m = slope, b = y-intercept

Inequalities

Key Rules

  • You can add or subtract the same value from both sides
  • You can multiply or divide by a positive number on both sides
  • IMPORTANT: When multiplying or dividing by a negative number, flip the inequality sign!

Example:

Solve: -3x + 5 > 14

Solution:

  • Subtract 5 from both sides: -3x > 9
  • Divide by -3 and flip the sign: x < -3
  • Answer: x < -3

Functions

Function Notation

f(x) represents a function where x is the input and f(x) is the output.

If f(x) = 2x + 3, then f(5) = 2(5) + 3 = 13

Common Function Types:

  • Linear: f(x) = mx + b (straight line)
  • Quadratic: f(x) = ax2 + bx + c (parabola)
  • Absolute Value: f(x) = |x| (V-shaped)
  • Exponential: f(x) = ax (rapid growth/decay)

Ready to Practice Algebra?

Apply these algebraic concepts with targeted practice questions.

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