Learning Target:
Begin a focus on systems of equations and will
examine the meaning of points of intersection.
Focus Questions
What does the y-axis represent?
What does the x-axis represent?
What would a given point represent in the context of the problem?
C
Pencils in the Middle
In Section 4.1, you graphed lines and curves that represented tile patterns. But what happens
when you graph two lines at the same time? What can you learn? Today you will use data,
graphs, and rules to examine what happens when two lines or curves intersect.
5-23. The Iditarod Trail Sled Dog Race is famous for its incredible length and
its use of dogs. The sled drivers, known as mushers, start their dog sleds at
Fairbanks, Alaska and ride through the snow for several days until they reach
Nome, Alaska. Along the route, there are stations where the competitors
check in, so data is kept on the progress of each team.
Joyla and her team of dogs have made it through the first five checkpoints. Her buddy Evie left Nome (the finish line) on the day the race started in an
effort to meet Joyla and offer encouragement. Evie traveled along the route
toward the racers on her snowmobile. The progress of each person is shown
on the graph that follows.
Your Task: With your team, analyze the data on the graph. Consider the questions below as you work. Be prepared to
defend your results.
• Which data represents Evie? Which
represents Joyla? How can you
tell?
• When did Evie meet Joyla?
• How long was the race? How can
you tell?
• Who traveled faster? Explain how
you know.
• Approximately how long did it take
Joyla to finish the race? How did
you find your answer?
• Which data represents Evie?
Which represents Joyla?
How can you tell?
• When did Evie meet Joyla?
• How long was the race? How
can you tell?
• Who traveled faster? Explain
how you know.
• Approximately how long did it take Joyla to finish the race? How did you find your
answer?
C
Think, Ink, Pair, Share
5-24. The point where two lines (or curves) cross is called a point of
intersection. Two or more lines (or curves) are called a system of equations. When you work with data, points of intersection can be meaningful, as you saw
in the last problem.
a. On graph paper, graph
y = 3x − 4 and y = −2x + 6
on the same set of axes.
b. Find the point of
intersection of these two
lines and label the point with
its coordinates; that is, write
it in the form (x, y). c. What is the significance of this point for the two rules in part (a)? Vocab
Tool Kit
pg. 42
System of Equations A system of equations is a set of equations with the same variables. Point of Intersection a point that the graphs of two equations have in common. For example, (3, 4)
is a point of intersection of the two graphs shown below. The ordered pair
representing a point of intersection gives a solution to the equations of each of
the graphs.
Solution to a System of Equations Solving a system of equations means finding one or more solutions that make
each of the equations in the system true. A solution to a system of equations
gives a point of intersection of the graphs of the equations in the system. Solutions to a Linear System of Equations A linear system of equations may have:
one solution (the point where the lines intersect)
no solution (because the lines never intersect)
infinitely many solutions (because the lines overlap)
Solving Equations with Fractions
(also known as Fraction Busters)
This equation would be much easier to solve if it had no fractions. Therefore, the first
goal is to find an equivalent equation that has no fractions.
To eliminate the denominators, multiply both sides of the equation by the common
denominator. In this example, the lowest common denominator is 15, so multiplying both
sides of the equation by 15 eliminates the fractions. 5x + 3x = 30
8x = 30
**NOTE:
Another approach is to multiply
both sides of the equation by one
denominator and then by the other.
1.25 + 0.75 = 2
Either way, the result is an equivalent
equation without fractions:
The number used to eliminate the denominators is called a Fraction Buster. Fraction Buster worksheet