Chapter 1
Linear Equations and Graphing
Course: Gre
Subject: Gre Quantitative Reasoning
Chapter: Linear Equations And Graphing
What Are Linear Equations?
A linear equation is a mathematical statement where every variable has an exponent of exactly 1. When graphed on a coordinate plane, a linear equation always produces a straight line.
The standard form of a linear equation is: ax + by + c = 0
Where:
- a, b, and c are integers
- a and b cannot both be zero at the same time
- x and y are the variables
Key Features of Linear Equations
Every linear equation has these important characteristics:
Slope (m): The rate of change, calculated as:
- m = change in y / change in x
- m = (y₂ - y₁)/(x₂ - x₁)
x-intercept: The point where the line crosses the x-axis (coordinates: x, 0)
y-intercept: The point where the line crosses the y-axis (coordinates: 0, y)
Example:
In the equation 2x + y - 3 = 0:
- Slope (m) = 2
- x-intercept = -3 (when y = 0)
- y-intercept = 3 (when x = 0)
Three Common Forms of Linear Equations
Linear equations can be written in different forms, each useful for specific situations:
| Form | Equation | Best Used For | | Standard Form | ax + by + c = 0 | Solving systems of equations | | Slope-Intercept Form | y = mx + b | Graphing and identifying slope/intercept | | Point-Slope Form | (y - y₁) = m(x - x₁) | When you know one point and the slope |
How to Write Linear Equations
The process of writing a linear equation in slope-intercept form (y = mx + b) follows three simple steps:
- Find the slope (m)
- Find the y-intercept (b)
- Substitute both values into y = mx + b
Let's explore different scenarios for writing linear equations.
Scenario 1: Given Slope and Y-Intercept
This is the simplest case. If you're told the slope is 2 and the y-intercept is 3:
- m = 2
- b = 3
- Equation: y = 2x + 3
Scenario 2: Given Two Points
When you know two points on the line, follow these steps:
Example: Find the equation of the line through points (1, 1) and (3, 2)
Step 1: Find the slope m = (y₂ - y₁)/(x₂ - x₁) = (2 - 1)/(3 - 1) = 1/2
Step 2: Find the y-intercept Use either point in y = mx + b Using (1, 1): 1 = (1/2)(1) + b 1 = 1/2 + b b = 1/2
Step 3: Write the equation y = (1/2)x + 1/2
Scenario 3: Given a Graph
When working from a graph:
Example: Find the equation of the line shown in a graph
Step 1: Find the slope Pick two points on the line, such as (-2, -6) and (2, 2) m = (2 - (-6))/(2 - (-2)) = 8/4 = 2
Step 2: Find the y-intercept Look where the line crosses the y-axis: at -2, so b = -2
Step 3: Write the equation y = 2x - 2
Scenario 4: Given a Point and a Parallel Line
When a line is parallel to another line, they have the same slope.
Example: Find the equation of the line parallel to y = 2x + 3 that passes through (2, 2)
Step 1: Find the slope Parallel lines have equal slopes, so m = 2
Step 2: Find the y-intercept Substitute the point (2, 2) into y = 2x + b 2 = 2(2) + b 2 = 4 + b b = -2
Step 3: Write the equation y = 2x - 2
Scenario 5: Given a Point and a Perpendicular Line
When a line is perpendicular to another line, their slopes are opposite reciprocals.
Rule: m₁ × m₂ = -1
Example: Find the equation of the line perpendicular to y = 2x + 3 that passes through (-2, 2)
Step 1: Find the slope The given line has slope m₁ = 2 For perpendicular lines: 2 × m₂ = -1 m₂ = -1/2
Step 2: Find the y-intercept Substitute the point (-2, 2) into y = (-1/2)x + b 2 = (-1/2)(-2) + b 2 = 1 + b b = 1
Step 3: Write the equation y = (-1/2)x + 1
Understanding Parallel and Perpendicular Lines
Parallel Lines
- Look like train tracks that never intersect
- Have the exact same slope
- Example: y = 3x + 2 and y = 3x - 5 are parallel
Perpendicular Lines
- Intersect at right angles (90°)
- Have slopes that are opposite reciprocals
- Opposite: change the sign (positive ↔ negative)
- Reciprocal: flip the fraction (a/b ↔ b/a)
- Example: If one line has slope 1/4, a perpendicular line has slope -4/1 = -4
- Important: You must do BOTH opposite AND reciprocal!
Common Mistakes to Avoid
When working with perpendicular lines, students often:
- Only make the slope negative but forget to flip it
- Only flip the fraction but forget to change the sign
- Remember: It's BOTH operations!
Summary
To write any linear equation:
- Always find the slope (m) first
- Then find the y-intercept (b) by substituting a known point
- Finally, write the equation as y = mx + b
How you find m depends on what information you're given:
- Two points: Use the slope formula
- Parallel line: Take the same slope
- Perpendicular line: Take the opposite reciprocal
- Graph: Pick two points and calculate