Chapter 1
Linear Equations and Graphing
Course: Gre
Subject: Gre Quantitative Reasoning
Chapter: Linear Equations And Graphing
What is a Linear Equation?
A linear equation is a mathematical statement that contains an equals sign (=) where all variables have an exponent of 1. This means no variable is squared, cubed, or has any exponent higher than 1. The word "linear" relates to "line" because when you graph a linear equation with two variables, it always forms a straight line.
Examples of Linear Equations:
- 2x - 3 = 5
- y = 9x - 3
- 4x + 5y = 46
- 3x - 2y + 5z = 36
Examples of NON-Linear Equations:
- x² + 4x - 1 = 0 (x is squared)
- 2√y = x (y is under a square root)
- y³/5 = 9 (y is cubed)
Real-World Applications of Linear Equations
Linear equations appear frequently in everyday situations:
Example 1: Distance and Speed The formula d = rt (distance = rate × time) is linear. If a car travels 120 miles at 30 miles per hour, we can find the time: 120 = 30t, so t = 4 hours.
Example 2: Temperature Conversion Converting Fahrenheit to Celsius uses (F - 32) × 5/9 = C. For 87°F: (87 - 32) × 5/9 = 30.556°C.
Example 3: Leaking Water A bucket with 128 ounces leaking at 10 ounces per minute follows A = 128 - 10t. After 5 minutes: A = 128 - 50 = 78 ounces.
Example 4: Polygon Angles The sum of interior angles in a polygon is (n - 2) × 180 = A. For a triangle (n=3): (3-2) × 180 = 180 degrees.
Standard Forms of Linear Equations (Two Variables)
Linear equations with two variables can be written in several standard forms, each useful for different purposes:
- General Form
ax + by + c = 0
- a, b, and c are real numbers
- Useful for solving systems of equations
- Slope-Intercept Form
y = mx + b
- m = slope (how steep the line is)
- b = y-intercept (where the line crosses the y-axis)
- Best for graphing
- Point-Slope Form
(y - y₀) = m(x - x₀)
- (x₀, y₀) is a point on the line
- m = slope
- Best when you know one point and the slope
- Two-Point Form
(y - y₁) = [(y₂ - y₁)/(x₂ - x₁)] (x - x₁)
- (x₁, y₁) and (x₂, y₂) are two points on the line
- The fraction in brackets calculates the slope
- Best when you know two points but not the slope
Linear Equations as Functions
When written in slope-intercept form, a linear equation can be expressed as a function: f(x) = mx + b
Special Linear Functions:
Identity Function: f(x) = x
- Slope = 1, y-intercept = 0
- A line at 45° angle through the origin
Constant Function: f(x) = c
- Horizontal line
- No x variable means y never changes
Solving Linear Equations
Solving One Variable Equations
Isolate the variable using inverse operations: 24x - 9 = 279 24x = 288 x = 12
Solving Systems of Two Equations
Methods include substitution (replacing one variable) and elimination (adding equations to cancel a variable).
Building Linear Equations from Real Situations
If you buy a car with 27,000 miles and drive 12,000 miles each year, the pattern is: y = 12,000x + 27,000
- x = years owned
- y = total miles
- Slope (12,000) = miles driven per year
- y-intercept (27,000) = starting miles
To find when the car reaches 250,000 miles: 250,000 = 12,000x + 27,000 223,000 = 12,000x x ≈ 18.6 years
Key Takeaways
- Linear equations have variables with exponent 1 only
- They graph as straight lines
- The slope (m) tells you the rate of change
- The y-intercept (b) tells you the starting value
- Different forms serve different purposes but represent the same line
- Linear equations help model and predict real-world patterns