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
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

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
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
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
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
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
 x1  x2 y1  y2 
M 
,

2 
 2
Examples

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Find the midpoint of the line segment with
the given endpoints:
(5, 2) and (-1, 8)
(4, -3) and (-1, 3)
About Slope
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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 ( x , y )
1 1
The slope of the line
through the points is
given by
( x2 , y2 )
y2  y1
m
x2  x1
Find the slope of the line passing
through the given points
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(-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
y  mx  b

Where m represents
the slope of the line and
(0, b) is the y-intercept.
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
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Through (-2, -3); m = 5/4
Through (5, 3); m = 0
Through (-4, 1); undefined slope
About Parallel and Perpendicular Lines
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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?
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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:
y  y1  m( x  x1 )
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)
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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
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
Find the equation of the line, in slopeintercept 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