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Graphing Linear Relationships study guide and practice problems
LINEAR RELATIONSHIP: Two properties that are related to each other
such that, when plotted on a graph, the points form a straight line.
The linear relationship is described by the equation that satisfies all points
on that line.
WHAT IS THIS EQUATION?
Y = mX + a
“m” is the slope
“a” is the Y-intercept
So, what is the SLOPE of a line?
SLOPE = change in Y = rise
change in X run
What is the Y-intercept of a line?
The Y-intercept is value of Y when X is equal to zero.
This point is the point at which the line crosses the Yaxis.
Examples:
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I. CONSIDER the relationship between sets of numbers such that one
number is always 4 times the second number. This relationship is a
LINEAR relationship. Therefore, it can be described using the equation
above.
Let us call the first number “Y”.
Therefore, since we know that “Y” is always four times as large as the
second number, “X”, the relationship between these two sets of numbers is:
Y = 4X
This is the equation for a line with a slope of 4 and a Y-intercept of zero.
II. CONSIDER the relationship between sets of numbers such that one
number is always 3 greater than 4 times the second number. This
relationship is also a LINEAR relationship. Therefore, it can be described
using the equation above.
This time we know that:
Y = 4X + 3
This is the equation for a line with a slope of 4 and a Y-intercept of 3.
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III. Practice problems
A. Find the slope and y-intercept for EACH of the following lines:
a. y = 3x + 15
b. y = 27 – 4x
c. 5y = x + 10
d. 3y = 27 – 6x
B. Write the equation of the line that satisfies the following:
a. A line with a slope of -1 and a y-intercept of (0, 15)
b. A line that passes through (5, 5) and (0, 3)
c. A line that passes through (16, 2) and (8, 1)
d. A line that passes through (3, 5) and has a slope of -1
III. Practice problems SOLUTIONS
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A. Find the slope and y-intercept for EACH of the following lines:
a. y = 3x + 15
y = mx + b; m = 3 and b = 15
Therefore, slope = 3 and y-intercept = (0, 15)
b. y = 27 – 4x
rewrite in the form of y = mx + b
y = -4x + 27
y = mx + b; m = -4 and b = 27
Therefore, slope = -4 and y-intercept = (0, 27)
c. 5y = x + 10
rewrite in the form of y = mx + b
y = 1/5x + 2
y = mx + b; m = 1/5 and b = 2
Therefore, slope = 1/5 and y-intercept = (0, 2)
d. 3y = 27 – 6x
rewrite in the form of y = mx + b
y = -2x + 9
y = mx + b; m = -2 and b = 9
Therefore, slope = -2 and y-intercept = (0, 9)
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B. Write the equation of the line that satisfies the following:
a. A line with a slope of -1 and a y-intercept of (0, 15)
using the equation of a line: y = mx + b
m = -1 and b = 15
Then, the equation of this line is:
y = -x + 15
b. A line that passes through (5, 5) and (0, 3)
using the equation of a line: y = mx + b
b = 3 but you must determine slope
change in y =
change in x =
(5 - 3) =
(5 – 0) =
slope = 2/5
Then, the equation of this line is:
y = 2/5x + 3
c. A line that passes through (16, 2) and (8, 1)
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using the equation of a line: y = mx + b
First, determine the slope:
change in y =
change in x =
(2 - 1) =
(16 – 8) =
1
8
slope = 1/8
Now determine the y-intercept by substituting (8,
1) into:
y = 1/8x + b
1 = (1/8)(8) + b
1=1+b
0=b
Therefore, the y=intercept = (0, 0)
The equation for this line is:
y = 1/8x
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d. A line that passes through (3, 5) and has a slope of -1
using the equation of a line: y = mx + b
m = -1
Now determine the y-intercept by substituting (3,
5) into:
y = -x + b
5 = -3 + b
5+3= b
8=b
Therefore, the y=intercept = (0, 8)
The equation for this line is:
y = -x + 8