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Linear Equation (Two Variables)

See how to write and graph a linear equation/inequality
(two variables).
18 examples and their solutions.

Slope between Two Points

Formula


m = y2 - y1x2 - x1

Example

Slope between (1, 1), (3, 5)
Solution

Example

Slope between (-4, 1), (1, -2)
Solution

Example

Slope between (-1, 2), (5, 2)
Solution

Example

Slope between (1, 2), (1, 3)
Solution

x-Intercept

Definition

The x-intercept is the intersecting point
of the graph and the x-axis.

Example

x-intercept of 3x + 4y = 12
Solution

y-Intercept

Definition

The y-intercept is the intersecting point
of the graph and the y-axis.

Example

y-intercept of 3x + 4y = 12
Solution

Slope-Intercept Form

Formula


y = mx + b

Example

Slope: 2, y-intercept: 1
Linear equation?
Solution

Example

Graph y = 2x + 1.
Solution

Example

5x + 3y = 6
→ Slope-intercept form
Solution

Point-Slope Form

Formula


y = m(x - x1) + y1

Example

Slope: 3
Point on the line: (1, 2)
Linear equation?
Solution

Two-Point Form

Formula


y = y2 - y1x2 - x1(x - x1) + y1
y = y2 - y1x2 - x1(x - x2) + y2
is also true.

Example

Points on the line: (2, 1), (5, 4)
Linear equation?
Solution

Intercept Form

Formula


xa + yb = 1

Example

x-intercept: -3, y-intercept: 2
Linear equation?
Solution

Linear Equations of Parallel Lines

Definition

Parallel Lines are the lines
that have the same slope
and have the different y-intercepts.
The red triangle symbols mean
these two lines are parallel.

Example

A line is parallel to y = 3x + 4
and it passes through (1, -2).
Linear equation?
Solution

Linear Equations of Perpendicular Lines

Formula



m1m2 = -1
Perpendicular Lines are the lines
that form a right angle:
90 degrees.

The red symbol is the right angle symbol.
If the slopes of two perpendicular lines are
m1, m2,
then m1m2 = -1.

Example

A line is perpendicular to y = 3x + 4
and it passes through (3, 1).
Linear equation?
Solution

Linear Inequality (Two Variables)

Example

Graph y > x + 2.
Solution

Example

Graph y ≤ 3x - 4.
Solution

Example

Graph y < 1 on a coordinate plane.
Solution

Example

Graph x ≥ -2 on a coordinate plane.
Solution