Graphs of Linear Equations
What is Graphs of Linear Equations?
An equation in which the highest power of the variable(s) is one, and which represents a straight line when plotted.
Key formula / rule: Slope Formula
Key points
- To plot the graph of a linear equation.
- To determine the slope and intercepts from an equation and a graph.
- To identify parallel and perpendicular lines from their equations.
- To interpret graphical representations of linear relationships.
Common exam trap
Confusing the x and y intercepts.
Definitions
- Term
Linear Equation
- Meaning
An equation in which the highest power of the variable(s) is one, and which represents a straight line when plotted.
- Term
Slope
- Meaning
A measure of the steepness and direction of a line; the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line.
- Term
Y-intercept
- Meaning
The point where a line crosses the y-axis; the value of y when x = 0.
- Term
X-intercept
- Meaning
The point where a line crosses the x-axis; the value of x when y = 0.
- Term
Parallel Lines
- Meaning
Two or more lines that have the same slope and never intersect.
- Term
Perpendicular Lines
- Meaning
Two lines that intersect at a right (90-°) angle; their slopes are negative reciprocals of each other.
Learning objectives
To plot the graph of a linear equation.
To determine the slope and intercepts from an equation and a graph.
To identify parallel and perpendicular lines from their equations.
To interpret graphical representations of linear relationships.
To solve systems of linear equations graphically.
Formulae
- Name
Slope Formula
- Note
Calculates the slope of a line given two points (x1, y1) and (x2, y2).
- Expression
m = (y2 - y1) / (x2 - x1)
- Name
Slope-Intercept Form
- Note
Represents a line where 'm' is the slope and 'c' is the y-intercept.
- Expression
y = mx + c
- Name
Point-Slope Form
- Note
Equation of a line passing through (x1, y1) with slope 'm'.
- Expression
y - y1 = m(x - x1)
- Name
General Form to Slope-Intercept Form
- Note
Here, slope m = -a/b and y-intercept = c/b.
- Expression
From ax + by = c, solve for y: y = (-a/b)x + (c/b)
Prerequisites
Basic algebra (variables, equations, solving for unknowns).
Coordinate geometry (Cartesian plane, plotting points).
Understanding of slope and intercepts.
Common mistakes
Confusing the x and y intercepts.
Incorrectly calculating the slope, especially with negative values or when the line is vertical.
Assuming all relationships are linear without checking.
Errors in plotting points or drawing the line.
Misinterpreting the graphical representation in the context of a word problem.
Keywords
Linear Equation
Graph
Straight Line
Slope
Intercept
Cartesian Plane
Coordinate Geometry
Parallel
Perpendicular
System of Equations
Practice preview
What is the equation of the line passing through the points (1, 2) and (3, 4)?…
medium
What is the x-intercept of the linear equation 2x + 3y = 6?…
easy
Does the point (2, 4) lie on the graph of the equation 3x - 2y = 0?…
easy