Transcript 8-10 Systems_of_Equations

Five-Minute Check (over Lesson 8–9)
Then/Now
New Vocabulary
Example 1: Solve by Graphing
Example 2: Standardized Test Example
Example 3: No Solution and Infinitely Many
Solutions
Example 4: Solve by Substitution
Concept Summary: Systems of Equations
Over Lesson 8–9
Which is an equation for the line of best fit for
the scatter plot where x is the years since 1998
and y is the number of visitors in thousands?
A. y = 3x + 15
B. y = x + 5
C. y = 3x – 15
D.
Over Lesson 8–9
Use the line of best fit, y = 3x + 15, to predict the
number of visitors in 2010.
A. 100,000
B. 80,000
C. 51,000
D. 21,000
Over Lesson 8–9
Caleb drew a scatter plot for the average number
of hours of homework y he received each year in
school x. A line of best fit passed through the
points (5, 1.5) and (8, 3.75). Predict the numbers
of hours of homework that Caleb can expect to
receive in 12th grade.
A. 4.75
B. 5.25
C. 6.75
D. 7.25
You have already solved linear equations by
graphing. (Lesson 8–2)
• Solve systems of linear equations by
graphing.
• Solve systems of linear equations by
substitution.
• system of equations
• substitution
Solve by Graphing
Solve the system of equations by graphing.
y = –x + 4
y = 2x + 1
Graph each line.
Solve by Graphing
The graphs appear to intersect at (1, 3). Check this by
replacing x with 1 and y with 3.
Check
y = –x + 4
?
3 = –1 + 4
1 = 1 
y = 2x + 1
?
3 = 2(1) + 1
3 = 3 
Answer: The solution of the system is (1, 3).
What is the solution of the system of equations?
y = –x + 3
y = 4x – 2
A. (2, 1)
B. (2, –1)
C. (1, 2)
D. (1, –2)
A. Catori and Mark each download songs. Mark
downloaded 4 times as many songs as Catori. Mark
also downloaded 6 more songs than Catori.
Write a system of equations to represent this
situation.
Let x represent Catori’s songs and y represent Mark’s
songs.
y = 4x
Mark downloaded 4 times as
many songs as Catori.
y = x+6
Mark downloaded 6 more songs
than Catori.
Answer: y = 4x and y = x + 6
B. Catori and Mark each download songs. Mark
downloaded 4 times as many songs as Catori. Mark
also downloaded 6 more songs than Catori.
Solve the system by graphing. Explain what the
solution means.
Graph the equations y = 4x and
y = x + 6 on the same coordinate grid.
The equations intersect at (2, 8).
Answer: So, the solution to the
system is x = 2 and y = 8. This means
that Catori downloaded 2 songs and
Mark downloaded 8 songs.
A. Andy ran twice as far as Ron and Andy ran
8 miles farther than Ron. Which system of
equations represents this situation?
A.
y = 2x
y=x–8
B.
y = 8x
y=x–2
C.
y = 2x
y=x+8
D.
y = 8x
y=x+2
B. Andy ran twice as far as Ron and Andy ran
8 miles farther than Ron. How far did each person
run?
A. Andy ran 10 miles.
Ron ran 5 miles.
B. Andy ran 12 miles.
Ron ran 6 miles.
C. Andy ran 16 miles.
Ron ran 8 miles.
D. Andy ran 18 miles.
Ron ran 9 miles.
No Solution and Infinitely Many Solutions
A. Solve the system of equations by graphing.
y = –x – 3
2x + 2y = –6
Both equations have the same
graph. Any ordered pair on the
graph will satisfy both
equations.
Answer: Therefore, there are infinitely many solutions
of this system of equations.
No Solution and Infinitely Many Solutions
B. Solve the system of equations by graphing.
y = 2x
y = 2x – 4
The graphs appear to be
parallel lines.
Answer: Since there is no coordinate pair that is a
solution to both equations, there is no solution of this
system of equations.
A. Solve the system of equations by graphing.
y=x+1
2y – 2x = 2
A. (0, 1)
B. (1, 2)
C. infinitely many solutions
D. no solution
B. Solve the system of equations by graphing.
y=x
y=x+1
A. (1, 1)
B. (1, 2)
C. infinitely many solutions
D. no solution
Solve by Substitution
Solve the system of equations by substitution.
y=7
y = 2x – 5
Replace y with 7 in the second equation.
y = 2x – 5
Write the second equation.
7 = 2x – 5
Replace y with 7.
12 = 2x
6=x
Add 5 to each side.
Solve for x.
Answer: The solution of this system of equations
is (6, 7).
Solve the system of equations by substitution.
y=8
y=x+4
A. (4, 8)
B. (–4, 0)
C. (8, 4)
D. (2, 8)