Five-Minute Check (over Chapter 2)
CCSS
Then/Now
New Vocabulary
Example 1: Solve by Using a Table
Example 2: Solve by Graphing
Example 3: Classify Systems
Concept Summary: Characteristics of Linear Systems
Key Concept: Substitution Method
Example 4: Real-World Example: Use the Substitution Method
Key Concept: Elimination Method
Example 5: Solve by Using Elimination
Example 6: Standardized Test Example: No Solution and Infinite Solutions
Concept Summary: Solving Systems of Equations
Over Chapter 2
Find the domain and range of the relation
{(–4, 1), (0, 0), (1, –4), (2, 0), (–2, 0)}. Determine
whether the relation is a function.
A. D = {–4, –2, 0, 1, 2},
R = {–4, 0,1}; yes
B. D = {0, 1, 2}, R = {0, 1}; yes
C. D = {–4, 0, 1},
R = {–4, –2, 0, 1, 2}; no
D. D = {–2, –4}; R = {–4, 0, 1}; yes
Over Chapter 2
Find the value of f(4) for f(x) = 8 – x – x2.
A. 28
B. 12
C. –12
D. –16
Over Chapter 2
Find the slope of the line that passes through (5, 7)
and (–1, 0).
A.
B.
C. 2
D. 7
Over Chapter 2
Write an equation in slope-intercept form for the
line that has x-intercept –3 and y-intercept 6.
A. y = –3x + 6
B. y = –3x – 6
C. y = 3x + 6
D. y = 2x + 6
Over Chapter 2
The Math Club is using the prediction equation
y = 1.25x + 10 to estimate the number of members it
will have, where x represents the number of years
the club has been in existence. About how many
members does the club expect to have in its fifth
year?
A. 15
B. 16
C. 17
D. 18
Over Chapter 2
Identify the type of function represented by the
equation y = 4x2 + 6.
A. absolute value
B. linear
C. piecewise-defined
D. quadratic
Mathematical Practices
2 Reason abstractly and quantitatively.
6 Attend to precision.
You graphed and solved linear equations.
• Solve systems of linear equations
graphically.
• Solve systems of linear equations
algebraically.
• break-even point
• independent
• system of equations
• dependent
• consistent
• substitution method
• inconsistent
• elimination method
Solve by Using a Table
Solve the system of equations by completing
a table.
x+y=3
–2x + y = –6
Solve for y in each equation.
x+y = 3
y = –x + 3
–2x + y = –6
y = 2x – 6
Solve by Using a Table
Use a table to find the solution that satisfies both
equations.
Answer: The solution to the system is (3, 0).
What is the solution of the system of equations?
x+y=2
x – 3y = –6
A. (1, 1)
B. (0, 2)
C. (2, 0)
D. (–4, 6)
Solve by Graphing
Solve the system of equations by graphing.
x – 2y = 0
x+y=6
Write each equation in slope-intercept form.
The graphs appear to
intersect at (4, 2).
Solve by Graphing
Check Substitute the coordinates into each equation.
x – 2y = 0
?
x+y =6
?
4 – 2(2) = 0
4+2 =6
0=0
6=6
Original equations
Replace x with 4
and y with 2.
Simplify.
Answer: The solution of the system is (4, 2).
Which graph shows the solution to the system of
equations below?
x + 3y = 7
x–y = 3
A.
C.
B.
D.
Classify Systems
A. Graph the system of equations and describe it as
consistent and independent, consistent and
dependent, or inconsistent.
x–y=5
x + 2y = –4
Write each equation in slope-intercept form.
Classify Systems
Answer:
The graphs of the equations intersect at (2, –3). Since
there is one solution to this system, this system is
consistent and independent.
Classify Systems
B. Graph the system of equations and describe it as
consistent and independent, consistent and
dependent, or inconsistent.
9x – 6y = –6
6x – 4y = –4
Write each equation in slope-intercept form.
Since the equations are equivalent, their graphs
are the same line.
Classify Systems
Answer:
Any ordered pair representing a point on that line will
satisfy both equations. So, there are infinitely many
solutions. This system is consistent and dependent.
Classify Systems
C. Graph the system of equations and describe it as
consistent and independent, consistent and
dependent, or inconsistent.
15x – 6y = 0
5x – 2y = 10
Write each equation in slope-intercept form.
Classify Systems
Answer:
The lines do not intersect. Their graphs are parallel lines.
So, there are no solutions that satisfy both equations.
This system is inconsistent.
Classify Systems
D. Graph the system of equations and describe it as
consistent and independent, consistent and
dependent, or inconsistent.
f(x) = –0.5x + 2
g(x) = –0.5x + 2
h(x) = 0.5x + 2
Classify Systems
Answer:
f(x) and g(x) are consistent and dependent. f(x) and h(x)
are consistent and independent. g(x) and h(x) are
consistent and independent.
A. Graph the system of
equations below. What type of
system of equations is shown?
x+y=5
2x = y – 5
A. consistent and independent
B. consistent and dependent
C. consistent
D. none of the above
B. Graph the system of
equations below. What type of
system of equations is shown?
x+y=3
2x = –2y + 6
A. consistent and independent
B. consistent and dependent
C. inconsistent
D. none of the above
C. Graph the system of
equations below. What type of
system of equations is shown?
y = 3x + 2
–6x + 2y = 10
A. consistent and independent
B. consistent and dependent
C. inconsistent
D. none of the above
D. Graph the system of equations below. Which
statement is not true?
f(x) = x + 2
g(x) = x + 4
A. f(x) and g(x) are consistent
and dependent.
B. f(x) and g(x) are inconsistent.
C. f(x) and h(x) are consistent
and independent.
D. g(x) and h(x) are consistent.
Use the Substitution Method
FURNITURE Lancaster Woodworkers Furniture
Store builds two types of wooden outdoor chairs. A
rocking chair sells for $265 and an Adirondack
chair with footstool sells for $320. The books show
that last month, the business earned $13,930 for
the 48 outdoor chairs sold. How many of each chair
were sold?
Understand
You are asked to find the number of each type of
chair sold.
Use the Substitution Method
Plan
Define variables and write the system of equations.
Let x represent the number of rocking chairs sold and
y represent the number of Adirondack chairs sold.
x + y = 48
265x + 320y = 13,930
The total number of chairs
sold was 48.
The total amount earned
was $13,930.
Use the Substitution Method
Solve one of the equations for one of the variables in
terms of the other. Since the coefficient of x is 1, solve
the first equation for x in terms of y.
x + y = 48
x = 48 – y
First equation
Subtract y from each side.
Use the Substitution Method
Solve Substitute 48 – y for x in the second equation.
265x + 320y = 13,930
265(48 – y) + 320y = 13,930
12,720 – 265y + 320y = 13,930
55y = 1210
y = 22
Second equation
Substitute 48 – y for x.
Distributive Property
Simplify.
Divide each side by 55.
Use the Substitution Method
Now find the value of x. Substitute the value for y into
either equation.
x + y = 48
x + 22 = 48
x = 26
First equation
Replace y with 22.
Subtract 22 from each side.
Answer: They sold 26 rocking chairs and 22
Adirondack chairs.
Use the Substitution Method
Check You can use a graphing calculator to check
this solution.
AMUSEMENT PARKS At Amy’s Amusement Park,
tickets sell for $24.50 for adults and $16.50 for
children. On Sunday, the amusement park made
$6405 from selling 330 tickets. How many of each
kind of ticket was sold?
A. 210 adult; 120 children
B. 120 adult; 210 children
C. 300 children; 30 adult
D. 300 children; 30 adult
Solve by Using Elimination
Use the elimination method to solve the system of
equations.
x + 2y = 10
x+y=6
In each equation, the coefficient of x is 1. If one equation
is subtracted from the other, the variable x will be
eliminated.
x + 2y = 10
(–)x + y = 6
y= 4
Subtract the equations.
Solve by Using Elimination
Now find x by substituting 4 for y in either original
equation.
x+y =6
Second equation
x+4 =6
Replace y with 4.
x =2
Subtract 4 from each side.
Answer: The solution is (2, 4).
Use the elimination method to solve the system
of equations. What is the solution to the system?
x + 3y = 5
x + 5y = –3
A. (2, –1)
B. (17, –4)
C. (2, 1)
D. no solution
No Solution and Infinite
Solutions
Solve the system of equations.
2x + 3y = 12
5x – 2y = 11
A. (2, 3)
C. (0, 5.5)
B. (6, 0)
D. (3, 2)
Read the Test Item
You are given a system of two linear equations and are
asked to find the solution.
No Solution and Infinite
Solutions
Solve the Test Item
Multiply the first equation by 2 and the second equation
by 3. Then add the equations to eliminate the y variable.
2x + 3y = 12
Multiply by 2.
5x – 2y = 11
4x + 6y = 24
(+)15x – 6y = 33
19x
= 57
Multiply by 3.
x=3
No Solution and Infinite
Solutions
Replace x with 3 and solve for y.
2x + 3y = 12
2(3) + 3y = 12
6 + 3y = 12
3y = 6
y=2
First equation
Replace x with 3.
Multiply.
Subtract 6 from each side.
Divide each side by 3.
Answer: The solution is (3, 2). The correct
answer is D.
Solve the system of equations.
x + 3y = 7
2x + 5y = 10
A.
B. (1, 2)
C. (–5, 4)
D. no solution