GRAPHING LINEAR EQUATIONS
COMMON MISTAKES
1
10/20/2009
Graphing-Coordinate System and Plotting
Points
How to Plot Points
The grid containing the x and y axes is
called the Cartesian Coordinate
Plane.
Points are plotted by using horizontal
and vertical distances from the
starting point called the origin
(where the x and y axes intersecthas the coordinates (0,0) ).
A point’s coordinates are labeled (x, y)
where x = distance right or left on the
x-axis and y = distance up or down
from the x-axis.
To graph the point, start at the origin,
(0,0), go the x distance on the x-axis
and then from that location, go the y
distance above or below your mark.
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2
Common Mistakes
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Confusing the x- and ydistances/directions.
Plotting A(3, 2), B(-1, 5),
C(-4, 0), D(2, -3) and E(5, -2)
would look like…
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B
C
E
A
D
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Graphing-Understanding Slope
How Slope affects the
graph’s direction
Recall: Slope, m, relates
the ‘slant’ of a line…
m > 0 slant: upward
m = 0 slant: horizontal
m < 0 slant: downward
m = undefined : vertical
Equations use slope, m, in
their formats (i.e. SlopeIntercept, and Point-Slope)
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3
Common Mistakes
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Incorrectly identifying slope or
graphing it.
Not realizing slope, m, really is
m = vertical change
horizontal change
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Example 1:

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y = 3x + 2
The slope is 3 NOT 3x.
Example 2:
8x + 3y = 11
Solving for y finds
the slope is − 8 .
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10/20/2009
Graphing-Slope (continued)
Identifying Slope to Graph
a Linear Function
Slope
has many definitions.
y −y
m = x − x for 2 points
m = rise = dy where the
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2
1
2
1
run
dx
d means ‘change in the
x distance or change in the
y’ distances.
Common Mistakes
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Example 1: Identify the slope in
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Incorrect: The slope is 3x.
Correct:
The slope is 3.
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Graphing requires at least
one point and the slope or
two points.
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Incorrectly identifying slope or graphing it.
Not realizing slope, m, really is
m = vertical change
horizontal change
y = 3x − 2
Example 2: Find the slope: 8x + 3y = 11
Incorrect: The slope is 8
Correct: Solving for y gives the slope, m,
as
−8
m=
3
4
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Graphing-Slope (continued)
Graphing a Linear
Function using it’s Slope
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
The form y= mx + b, known as the
Slope-Intercept Form, is easily graphed.
Step 1: Start by solving the equation into
y=mx+b form, where m= slope (put into
fraction form and b=y-intercept.)
Step 2: Plot (0, b)
Step 3: Use the slope m to find another
point..
m>0, m=0, m<0, or m is undefined.
The numerator is the up/down (vertical
distance) and the denominator directs the
distance left/right (Always associate the
negative numbers with the numerator).
Step 4: Connect the points and complete
the line.
Common Mistakes


Incorrectly identifying slope or graphing it.
Not realizing slope, m, really is
m = vertical change
horizontal change

Example 1: Identify the slope in

Incorrect: The slope is 3x.
Correct:
The slope is 3.

y = 3x − 2
Example 2: Find the slope: 8x + 3y = 11
Incorrect: The slope is 8
Correct: Solving for y gives the slope, m,
as
−8
m=
3
5
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10/20/2009
Writing the Equation of a Line
How to Write the
Equation of a Line
Common Mistakes
Lines are written in three basic equation
forms…
Slope-Intercept Form: y = mx +b
Standard Form: Ax + By = C
Point-Slope Form: y - y 1 = m(x- x )
for some point ( x1 , y 1 ) and slope
m.
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1
To Write the Equation of the line…
Step 1: Note the Form of the Equation.
Step 2: Calculate the Slope (using either
the definition or applying
the given value).
Step 3: Substitute values into the equation
appropriately.
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6
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Miscalculating slope by incorrectly
substituting into the defined expression.
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Not correctly graphing the line given by
the newly found equation.
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Example: Write the equation of the line,
in slope-intercept form, for
point (2, -5) and m= 7.
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Solution: Using y=mx + b, we substitute
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m = 7 and calculate b.
-5 = 7(2) + b
-19 = b so y= 7x - 19
10/20/2009
Graphing– Horizontal and Vertical Lines
How to Graph Horizontal
and Vertical Lines
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7
Horizontal Lines have the form y = a, with a
slope of 0, or
m = 0.
Graphing Horizontal Lines:
Step 1: Start at the origin (0, 0) and move
up/ down
to(0, a).
Step 2: Plot (0, a) and draw a horizontal line
through that point.
Common Mistakes
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Confusing which form is vertical
and which one is horizontal.
Graphs: y=5 and x= -2
x= -2
Slope is undefined
y=5
Slope =0
Vertical Lines have the form x = b, with an
undefined
slope.
Graphing Vertical Lines
Step 1: Start at the origin
(0, 0) and move right/left to (b, 0)
Step 2: Plot (b, 0) and draw
a vertical line through that point.
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Graphing– Parallel and Perpendicular Lines
How to Graph Parallel and
Perpendicular Lines
Parallel Lines are lines with the
same slope.
Perpendicular Lines intersect lines
at 90˚ (Right Angles) and have
slopes that are negative
reciprocals.
When graphing, use the
techniques of plotting the yintercept and then using the slope
from y=mx + b, where m= slope
and b= y-intercept (0,b)
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8
Common Mistakes
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Forgetting or confusing the slope relationships
between parallel and perpendicular
Example: Write the equation of the lines
parallel and perpendicular to the line
y = 3 x +1 through (2, 4).
2
Solution: The point has values x=2 and y=4.

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Parallel: same slope, m= 3/2 .
Solving y=mx +b, where 4= 3 (2) + b
2
3
gives b = 1 so y= 2 x + 1.
Perpendicular: slopes that are negative
reciprocals, m= -2/3
Solving y = mx + b, where 4= − 2 (2) +b
3
gives b= − 16 so y = − 2 x + − 16 .
3
3
3
10/20/2009
Graphing- Plotting Points using the
Standard Form of an Equation
How to Graph an Equation
in Standard Form
Recall: Standard Form of a
Linear Equation is…
Ax + By =C.
Solving for y gives: y = − A x + C
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B
B
B
Graphing the line is done by first
plotting the y-intercept, or b, as the
coordinate (0, b); then use the
slope to plot a second point.
Connect the two points and
complete the line.
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B
When compared to y =mx + b,
Slope m= − A , y-intercept b = C .
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Common Mistakes
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Incorrectly solving for slope. (Apply the
formula: y = − A x + C ).
B
B
A=2
Example: Graph : 2x + 5y = 10 B =5
C=10
Solution: Where m = − 2 and b = 2 ,
5
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Plot (0, 2) and then use the slope by going
DOWN 2 and RIGHT 5 to find a second
point. Connect the line joining the points.
.
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2x+5y=10
10/20/2009
Graphing-Applying the Y-Intercept Form
How to calculate and
make use of Intercepts
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The y-intercept (0, b) can be calculated
by setting x=0 and solving the equation
y=mx+b for y.
The x-intercept (x, 0) can be calculated
by letting y=0 and solving y = mx + b for
x.
Using intercepts to graph lines in the
Standard Form Ax + By =C can be done
easily by a pattern:
x-intercept: if y=0, then x = CA ,
y-intercept: if x =0, then y =C ,
B
Plot and connect the two Intercepts (x, 0)
and (0,y); then complete the line.
Common Mistakes
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Incorrectly plotting the intercepts
when graphing: plotting a point(s)
with the coordinates for x and y are
‘backwards’.
Example: Find the intercepts that
would be used to graph …
A=3,
B=-2,
C=12
Incorrect: (0, 12 ) and ( 12 , 0)
−2
3
are NOT the intercepts.
3x − 2y = 12

( ,0) and (0, 12 ).
Correct: 12
−2
3
10
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10/20/2009