Everything You Will Ever Need To Know About Linear Equations*

*Whether You Wanted To Know It Or Not!

Definition

A linear equation in standard form can be written in the form

Ax By C

Important Fact

The graph of every linear equation is a straight line.

The line may be horizontal, vertical, or slanted.

How To Graph A Linear Equation, Part 1

Make a table of values Choose values for x or y Substitute into the equation and solve for the

other variable Make ordered pairs out of the values in the

table Graph the points and draw a line through

them

Examples

y = x + 3 for x = 0, 1, 2, 3, 4 x + 3y = 5 for x = 0, 1 and y = 0, -1 y = (1/2)x + 6 for x = ?

How To Graph A Linear Equation, Part 2

Find the x-intercept by letting y = 0 and solving for x.

Then find the y-intercept by letting x = 0 and solving for y.

Graph the two points and draw a line through them.

Examples

5x + 2y = 10 x – 2y = -4

Midpoint of a Line Segment

The midpoint M of a line segment with endpoints (x1, y1) and (x2, y2) is given by

2

,2

2121 yyxxM

Examples

Find the midpoint of the line segment with the given endpoints:

(5, 2) and (-1, 8) (4, -3) and (-1, 3)

About Slope

The slope of a line is its steepness or slant. Positively-sloped lines are uphill from left to

right. Negatively-sloped lines are downhill from left

to right. Horizontal lines have zero slope. Vertical lines have undefined slope.

Important Formula

Given two points The slope of the line

through the points is given by

1 1( , )x y 2 2( , )x y

2 1

2 1

y ym

x x

Find the slope of the line passing through the given points

(-4, 1) and (-3, 4) (-6, 3) and (2, 3) (-3, 1) and (6, -2) (4, -1) and (4, 3)

Definition

A linear equation in slope-intercept form can be written in the form

Where m represents the slope of the line and (0, b) is the y-intercept.

y mx b

How To Graph A Linear Equation, Part 2

Write the equation in slope-intercept form. Graph the y-intercept (0, b) Use the slope m to “rise” and “run” from the

y-intercept to another point on the graph.

About Horizontal and Vertical Lines

The equation of every horizontal line is y = k, where k is some number. Horizontal lines have zero slope.

The equation of every vertical line is x = h, where h is some number. Vertical lines have undefined slope.

Examples

Find the slope of the line and sketch the graph:

x + 3y = -6

4x – y = 4

y = -3x

x + 2 = 0

y = -4

Graph each line described

Through (-2, -3); m = 5/4 Through (5, 3); m = 0 Through (-4, 1); undefined slope

About Parallel and Perpendicular Lines

If two lines in the plane are parallel (do not intersect), the lines have equal slopes.

If two lines in the plane are perpendicular (meet at a 90-degree angle), the lines have “opposite-reciprocal” slopes.

Parallel, Perpendicular or Neither?

2x + 5y = -7 and 5x – 2y = 1 3x = y and 2y – 6x = 5 2x + 5y = -8 and 6 + 2x = 5y

How To Find The Equation of a Line

If you are given the slope of the line and a point on the line, use point-slope form:

1 1( )y y m x x

Examples

Write the equation, in slope-intercept form, of the line that passes through the given point with the given slope:

Through (7, -2); slope ¼

Through (2, 0); slope -5

Through (-4, -2); slope 0

Through (-2, 8); undefined slope

How to Find the Equation of a Line (continued)

If you are given two points on the line, use the slope formula to find the slope of the line. Then use point-slope form.

Examples

Write the equation, in standard form, of the line passing through the given points:

(-2, 5) and (-8, 1)

(5, -2) and (-3, 14)

How to Find the Equation of a Line (continued)

If you are told that your line is parallel or perpendicular to a given line, find the slope of that line.

If the lines are parallel, use that slope. If the lines are perpendicular, use the

opposite reciprocal of that slope. Use point-slope form to find your equation.

Examples

Find the equation of the line, in slope-intercept form, that passes through (4, 1) and is parallel to 2x + 5y = 10

Find the equation of the line, in standard form, that passes through (2, -7) and is perpendicular to 5x + 2y = 18