Elementary Algebra
Test #3 Review
Section 3.1: Reading Graphs: Linear Equations in Two Variables
Graph: c vs. n
 cost goes on vertical axis (dependent variable)
 number goes on horizontal axis (independent variable)
Practice:
1. Graph y = 2x – 9.
2. Graph 5x + 2y = 20.
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Section 3.2: Graphing Linear Equations in Two Variables
Big Idea: There are several techniques for graphing linear equations and several shortcuts to look out for.
Technique #1 for graphing a line: BRUTE FORCE
1. Solve the linear equation for the dependent variable (i.e., y)
2. Pick some random values of the independent variable (i.e., x), and calculate y; make a table of the
values.
3. Graph the points and connect them with a straight line.
Note: you really only need to calculate 2 points, but calculating more helps reveal any mistakes you might have
made.
Practice: Graph 3x – 6y = 18.
Solve for dependent
variable
Make table of values
x
y
Graph points
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Technique #2 for graphing a line: USE THE INTERCEPTS (uses the fact that you only need 2 points to
draw a line)
1. Find the x-intercept (which is where the line crosses the x-axis) by setting y = 0 and solving for x.
2. Find the y-intercept (which is where the line crosses the y-axis) by setting x = 0 and solving for y.
3. Graph the points and connect them with a straight line.
Practice: Graph 4x + 7y = 28.
Find x-intercept
Find y-intercept
Graph points
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Technique #3 for graphing a line: RECOGNIZE SPECIAL CASES
1. Horizontal line: The graph of the equation y = k is a horizontal line with y-intercept (0, y) and no
x-intercept.
2. Vertical line: The graph of the equation x = k is a vertical line with x-intercept (x, 0) and no
y-intercept.
3. Line through the Origin: The graph of the equation Ax + By = 0 is a slanted line through the origin
(0, 0). To graph the line, you’ll need a second point, so pick a random value for x, calculate y, then
graph.
Practice:
Graph y = -2.
Graph x = 4.
Graph 12x + 18y = 0.
Recognize special case
Graph line
Section 3.3: The Slope of a Line
Big Idea: Equal increments of x result in equal increments of y for a line. The ratio of these two increments is
called the slope. The slope is a measure of the “steepness” and “direction” of a line. We’ll use this later for a
fourth technique for graphing lines.
Definition: Slope of a Line
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The slope of a line is defined: slope 
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vertical change in y (rise)
horizontal change in x (run)
Practice:
1. Find the slope of the line connecting the points (1, 2) and (5, 4)
2. Find the slope of the line described by the equation 4x – 12y = 24.
Note: The slope of a line is the same between any two points on the line.
Slope Formula:
The slope of the line through the points (x1, y1) and (x2, y2) is m 
y2  y1
x2  x1
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Practice:
3. Find the slope of the line connecting the points (-3, -1) and (2, 5) using the slope formula.
4. Solve the equation 4x – 12y = 24 for the variable y, then relate the slope of the line to its equation.
Finding the Slope of a Line from its Equation:
 Solve the linear equation for the dependent variable (i.e., y)
 The coefficient of the x term is the slope.
Practice:
5. Find the slope of the line described by the equation 2x + 6y = 12. Also graph the line.
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Technique #4 for graphing a line: SLOPE INTERCEPT METHOD
 Solve the linear equation for the dependent variable (i.e., y)
 The constant term is the y-intercept. Plot it.
 The coefficient of the x term is the slope. Use it to count up and over (or down and backward) to plot
more points on the line.
Practice:
6. Graph 21x – 7y = 28 using the slope intercept method.
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Facts about slope:
1. A steep line has a large numerical slope.
2. A not so steep line has a small numerical slope.
3. The “dividing line” is the 45 line, which has a slope of m = 1.
4. Lines that go “uphill” (or are increasing as you read the graph from left to right) have a positive slope.
5. Lines that go “downhill” (or are decreasing as you read the graph from left to right) have a negative
slope.
6. Horizontal lines have a slope of m = 0.
7. Vertical lines have an undefined (or infinite) slope.
8. Parallel lines have the same slope.
9. Perpendicular lines have slopes whose product is -1.
Section 3.4: Equations of a Line
Big Idea: This section concludes our study of techniques for graphing lines, and summarizes some different
ways of writing the equation of a line.
Technique #4 for graphing a line: SLOPE INTERCEPT METHOD
4. Solve the linear equation for the dependent variable (i.e., y) and simplify. Your equation will be in
“slope-intercept” form: y = mx + b.
5. The constant term (b) is the “y-intercept”. Plot it (i.e., (0, b)).
6. The coefficient of the x term (m) is the slope. Use it to count up and over to plot more points on the line.
Examples:
1. Graph the line 2x + 3y = 12.
2. Find the equation of the line with slope 2 and y-intercept (0, 8), then graph it.
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3. Find the equation of the line with slope -½ and y-intercept (0, 3), then graph it.
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Technique #5 for graphing a line: POINT SLOPE METHOD This method is used when a slope and a point
OR two points are given.
1. Graph the line using given data.
2. Find the equation for the line using the “point-slope form” for a line: y – y1 = m(x – x1). Note: if you are
only given two points, then you’ll have to calculate the slope yourself first.
Examples:
1. Graph the line with slope -4 that passes through the point (-9, -5), and find its equation.
2. Graph the line that passes through the points (-4, 7) and (6,-3), and find its equation.
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3. Find the equation of the line that passes through the point (2, 2) and is parallel to the line
3x – 4y = 10, then graph it.
4. Find the equation of the line that passes through the point (2, 2) and is perpendicular to the line
3x – 4y = 10, then graph it.
Section 3.5: Graphing Linear Inequalities in Two Variables
Big Idea: This section looks at how to graph inequalities; the upshot is that you have to shade in one side of the
plane or the other.
Examples of Linear Inequalities:
7. Ax + By < C
8. Ax + By > C
9. Ax + By  C
10. Ax + By  C
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11. If I belong to a golf club with a flat fee of $200 for the year and where each round of golf costs $30 to
play, and I want to know the number of games n I can play if I want to spend less than c dollars on golf:
200  30n  c
To graph a linear inequality:
1. Graph the boundary line
a. If the inequality uses  or , then draw a solid line to show that the line itself satisfies the
inequality.
b. If the inequality uses just < or >, then draw a dashed line to show that the line does not satisfies
the inequality.
2. Shade the appropriate side.
a. Pick any point not on the line for a test point, then plug its x and y values into the inequality.
b. If the inequality is true for the test point, then shade in that side of the line.
c. If the inequality is false for the test point, then shade in the opposite side of the line.
Practice:
4. Graph the inequality 3x + 5y > 15.
5. Graph the inequality y < 4.
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6. Graph the inequality x  -3y.
7. Graph the inequality 200  30n  c .
Section 3.6: Introduction to Functions
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Big Idea: There are 4 main ways to specify a relation between numbers in mathematics: listing related pairs of
numbers as ordered pairs, drawing a graph of the relationship between the numbers, giving a verbal description
between related quantities, or specifying a calculation that relates numbers. This section focuses on the
calculation method, and the fact that when we give the calculation a name and make sure it doesn’t double up
its output values, then we call it a function.
Big Skill: You should be able to determine when a relation is a function, find the domain and range of a
function, and evaluate a function for given input values.
Math gets really exciting (!) when we start defining relationships between two quantities. For instance, if
we want to describe the relationship between the distance a car has traveled and the amount of time it has been
traveling, we could do it in many different ways:
1. Listing related numbers as ordered pairs:
a. After 1 hour, the car has traveled 65 miles  (1 hr, 65 mi)
b. After 2 hours, the car has traveled 130 miles  (2 hr, 130 mi)
c. After 3 hours, the car has traveled 195 miles  (3hr, 195 mi)
2. Drawing a graph of the relationship:
3. Give a verbal description:
a. The car travels 65 miles for every hour it has traveled.
4. Specify a calculation, but give it a name.
a. Let t represent the number of hours the car has been traveling.
b. Let d represent the distance traveled.
c. The fact that distance is a function of time can be written in function notation as d(t) = 65t.
Notice that the name of the calculation is d (for distance), and that after the name of the calculation, we put the
independent variable in parentheses so it is clear what the independent variable is. This is useful for functions
like f(x) = mx + b.
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Also notice that in each of the four ways of describing the relationship between time and distance, everything
boils down to a set of ordered pairs…





The set of all first components in all the ordered pairs of any relation (i.e., all the x’s) is called the
domain of the relation.
The set of all second components in all the ordered pairs of any relation (i.e., all the y’s) is called the
range of the relation.
Sometimes we represent the relation using a diagram with the domain and range in separate “blobs” and
arrows connecting related numbers between the “blobs”.
A function is a special type relationship where each element in the domain corresponds to exactly one
element in the range.
If a vertical line intersects the graph of a function at more than one point, then the graph is not the graph
of a function.
Practice: Identify the domain and range of the following relations, and then determine if the relation is a
function.
1. {(-2, -2), (-1, 1), (0, 2), (1, 1), (2, -1)}
2. {(-5, 2), (0, 2), (5, 6), (6, 5), (2, 0), (-5, 5)}
3.
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4.
5.
6.
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7. y = 2x – 4
8. y = x2 + 4
9. If f(x) = 6x – 2, compute f(2).
10. If f(x) = x2 + 4, compute f(-3).
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