1
Graphing Linear Equations
Graphing by Plotting Points
(Developmental Algebra)
The graph of a linear equation is a straight line. The equation is called a linear
equation when the degree of the polynomial function is 1. In order to graph a
straight line, a minimum of two points is required; however, it is recommended
that at least three points are found to ensure accuracy. To find points from any
given equation, simply choose values for the x or y coordinates and plug the
chosen values into the equation in order to find the missing values. You can
choose any number, but to keep the arithmetic simple, choose small values. To
find those values, make a table as follows.
Example:
Graph the equation y = 2x + 3
x
y
Ordered
Pair
-1
1
(-1, 1)
0
3
(0, 3)
1
5
(1, 5)
y
y
y
y
y
y
=
=
=
=
=
=
2(-1) + 3
1
2(0) + 3
3
2(1) + 3
2
Once the three points (ordered pairs) have been found, they can be
plotted on the rectangular coordinate system.
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Linear Equations in Two Variables
2
Example
Graph the equation 4x + y = 2
x
y
-1
6
(-1, 6)
4(-1) + y = 2
y=6
0
2
(0, 2)
4(0) + y = 2
y=2
1
-2
(1, -2)
y = 4(1)+ 1
y = -2
Other examples
Graph the equation x = -2
Graph the equation y = 5
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Linear Equations in Two Variables
3
Graphing Equations with Fractions
Method 1
To get rid of the fractions, find the L.C.D. of the equation and multiply each
term of the equation by the L.C.D.
Example:
Graph the equation of the line y =
2
3
x–1
The L.C.D. of the equation is 3
(3)y =(3)
3y =
6
3
2
3
x – (3)1
x–3
3y = 2x – 3
Now you can graph by finding and plotting the x and y-intercepts
x
y
Ordered
Pair
0
-1
(0, -1)
3
2
0
3
( , 0)
2
3y = 2(0) - 3
y = -1
3(0) = 2x - 3
x=
3
2
Method 2
Another method to avoid dealing with fractions is to pick multiples of the
L.C.D. for the variable whose coefficient is a fraction.
Graph the equation of the line y =
6
5
x–3
The L.C.D. of the equation is 5, so pick 5 and -5
Ordered y = 6 (-5) – 3
5
Pair
x
y
-5
-8
(-5, -8)
5
3
(5, 3)
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y = -8
y=
6
5
(5) – 3
Linear Equations in Two Variables
4
Graphing Linear Equations by Plotting the x and y intercepts
(All levels)
1. To find the x-intercept, set y equal to zero and solve for x.
2. To find the y-intercept, set x equal to zero and solve for y.
3. Plot the x and y-intercepts on the rectangular coordinates and draw a
line connecting the two points.
Example:
Graph the equation of the line y =5x + 6
x
y
0
6
(0, 6)
y = 5(0) + 6
y=6
−
6
5
0
6
0 = 5x + 6
(− , 0)
5
x = -6
5
6
To graph the fraction, convert it into a decimal. − is -1.2. This value is
5
between -1 and -2.
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Linear Equations in Two Variables
5
Forms of a Line
(Intermediate Algebra, College Algebra)
There are two forms that are used with linear equations:
Standard form: Ax +By = C, where A, B, and C are real numbers.
Slope-intercept form: y = mx + b, where m = slope and b = y-intercept. It is
important to remember that you can switch between the two forms just by
algebraically rearranging a problem.
Example:
Write 2x + 3y = 12 in the form y = mx + b
2x + 3y = 12
Solve for y
3y = -2x + 12
Use Addition Property of Equalities
3
3
y=−
2
3
x+
12
Use Multiplicative Property of Equalities
3
2
y=− x+4
3
Standard Form Ax + By = C
Standard Form is one of the two forms of a linear equation. The letters A, B,
and C represent integers. A, B, and C can’t be fractions and A has to be a
positive integer. The most simple way to graph an equation in standard form is
to find the x and y intercepts.
Example:
Find the intercepts of 2x + 4y = 12.
x
y
0
6
(0, 3)
2(0) + 4y = 12
y=3
6
0
(6, 0)
2x + 4(0) = 12
x=6
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Linear Equations in Two Variables
6
Slope-Intercept Form y = mx + b
The slope of the line is a number that describes the direction and
the steepness of the line and is calculated by finding the ratio of the "vertical
change" to the "horizontal change" between (any) two points on a line (rise over
run).
Slope-intercept form is the second form to represent a linear equation, where
m represents the slope and b the y-intercept. To graph a linear equation in this
form:
1. Plot the point (0, b) on the y-axis. This is the y intercept.
2. Use the slope to find a second point on the line; rise over run, or the
number of units up (or down) over the number of units to the right (or left)
that you need to move from the point (0, b). This is why the slope is always a
6
fraction. A slope of 6, for example,is the same as the fraction 1, so move six
units up from the point and one unit to the right. With a slope of 6, you may
also move six units down and one unit to the left, since
−6
−1
= 6.
Example
Graph the line y =
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−5
2
x +3
Linear Equations in Two Variables
7
Plot the point (0. 3)
Use the slope
−5
2
to find the next point
From (0, 3) go 5 units down (- sign) and then 2 units right (+sign)
Slope Formula
𝑦2 − 𝑦1
𝑚=
𝑥2 − 𝑥1
This formula allows you to find the slope of a line, so long as you have two
points that the line passes through. In the formula, m represents the slope. y2
and y1 represent the y-values of the points and x2 and x1 represent the x-values
of the points. To find the slope of the line:
1. Label the points.
2. Plug them into the formula.
Example:
What is the slope of the linear equation that passes through (1,-3) and
(-5, 4)?
(1,-3)
x1, y1
𝑚=
4−(−3)
𝑚=
7
(-5, 4)
x2, y2
−5−1
−6
=-
Label the points
Plug them into the formula
7
6
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Linear Equations in Two Variables
8
Point Slope Formula
y – y1 = m(x – x1)
As long as you have one point that passes through the line and the slope of the
line, you can find the equation of a line using the Point Slope Formula. With
this formula, you can find the equation of the line in either slope intercept form
or standard form.
To find the equation of the line in Slope Intercept Form:
1. Replace m with the given slope.
2. Replace x1 and y1 using the x, and y values of the given point.
3. Solve for y.
Example:
Find the equation of the line that passes through the point (-3, 5) with
slope of -4
y – 5 = m(x – (-3))
Replace x1 and y1
y – 5 = -4(x + 3)
Replace m with the slope
y – 5 = -4x – 12
Simplify R.H.S.
y = -4x -12 + 5
Use Addition Property of Equalities
y = -4x -7
To find the equation of the line in Standard Form:
1. Replace m with the given slope.
2. Replace x1 and y1 using the x and y values of the given point.
3. Get rid of any fractions, if the slope is given as a fraction.
4. Solve the equation, making sure that the x and y terms are on the
same side and that the constant term is on the other side of the equal
sign.
Example:
Find the equation of the line that passes through the point
(-5, -7) with slope of −
y – (-7) = m(x – (-5))
2
y – (-7) = − (x – (-5))
3
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2
3
Replace x1 and y1
Replace m with the slope
Simplify both sides
Linear Equations in Two Variables
9
y+7=−
2
3
(x + 5)
(3)y + (3)7 = (3) −
To get rid of the fraction multiply
2
3
every term of the equation by 3
(x + 5)
3y + 21 = -2(x + 5)
Simplify R.H.S.
3y + 21 = -2x - 10
The terms with a variable on one
side (the coefficient of x has to be
positive) and constant term on the
other side
3y +2x = -10 – 21
2x + 3y = - 31
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Linear Equations in Two Variables