Sophia Learning, College
Algebra, Milestone 3, WITH
ANSWERS
1
The graph of a linear function passes through the points
and .
Find the slope of this f... [Show More] unction.
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CONCEPT
Determining Slope
2
The prices for a loaf of bread and a gallon of milk for two
supermarkets are shown below.
Since we have two points from a linear function, we can
use the slope formula to find the slope of the line.
Now that the numbers are plugged in, evaluate the
subtraction in both the numerator and the denominator.
In the numerator, the difference in coordinates is
minus , or . In the denominator, the difference in
coordinates is minus , which is the same as plus
, or . The slope of the line is .
The slope is the difference in coordinates from the two
points divided by the difference in coordinates from the
same two points. When plugging in the values it is
important to be consistent with which coordinates are
subtracted in the calculations.
A B
Bread $2.20 $2.50
Milk $3.60 $3.40
Mary needs to buy bread and milk for her church picnic. At
Supermarket A, she would pay $44.20. At Supermarket B, she
would pay $44.70.
Which of the following system of equations represents this
situation?
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In general, the equation to represent the
total cost of buying bread and milk would
be the sum of the cost for bread and cost
for milk. To find the cost of bread and
milk, first define variables to represent the
amount of bread and milk.
Here, will represent bread, and will
represent milk. Both and will be
multiplied by their respective prices.
CONCEPT
Writing a System of Linear Equations
3
Select the correct slope and y-intercept for the following linear
equation:
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At Store A, a loaf of bread costs $2.20,
and a gallon of milk costs $3.60. We also
know that the total cost at Store A is
$44.20. So the total cost at Store A would
be expressed with this equation. We can
construct a similar equation for Store B.
At Store B, bread cost $2.50 per loaf, and
milk costs $3.40 per gallon. We also know
that the total cost at Store B is $44.70.
The total cost at Store B would be
expressed with this equation.
This is the system of equations to
represent the costs of bread and milk at
Store A and B.
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RATIONALE
CONCEPT
Forms of Linear Equations
4
Consider the region shaded in yellow.
Equations in the form allow us to easily
identify the slope and y-intercept. The slope is given by
the variable , and the y-intercept is given by the
variable .
The variable is the coefficient in front of x that
represents the slope. In the equation , the
coefficient in front of is , so is the slope.
The variable represents the y-coordinate of the yintercept.
In the equation , 8 will be the ycoordinate
of the y-intercept. Remember that the xcoordinate
of the y-intercept is always 0, so the yintercept
is .
Which inequality does the shaded region represent?
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Writing a linear inequality from a graph is
similar to writing a linear equation from a
graph. The biggest difference is that with
inequalities, a portion of the coordinate
plane is highlighted to represent the
solution region. Start by focusing on the
line that we see. Determine the yintercept
and slope so that we can write
an equation in slope-intercept form.
The graph intercepts the y-axis at (0,3) so
this is our intercept. To find the slope, we
can use this point and another point on
the line, for example, (-2,-1).
Plugging these two points into the slope
formula, we get a slope of 2. We can write
an equation in the form y = mx + b, where
m is the slope and b is the y-coordinate of
the y-intercept.
CONCEPT
Writing a Linear Inequality from a Graph
5
A line passes through the point and has slope of .
Find the equation of this line.
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Using the slope of 2 and the y-intercept of
3, we get the equation y = 2x + 3. Next,
we need to turn this equation into an
inequality. To do this, note two things: the
type of line used to graph the inequality,
and the highlighted solution region. The
graph has a solid line, which means that
the inequality symbol is either ≤ or ≥. To
decide between ≤ or ≥, determine whether
the region above or below the line is
highlighted. Since the region BELOW the
line is highlighted, the inequality symbol is
“less than or equal to”, ≤.
Because the line is solid and the region is
shaded below the line, the inequality y ≤
2x + 3 corresponds to this graph.
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CONCEPT
Writing a Linear Equation Using Slope and Points [Show Less]
