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Lesson Menu
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
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5-Minute Check 1
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
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5-Minute Check 2
Find the value of f(4) for f(x) 8 x x2.
A. 28 B. 12 C. 12 D. 16
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5-Minute Check 3
Find the slope of the line that passes through
(5, 7) and (1, 0).
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5-Minute Check 4
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
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5-Minute Check 6
Identify the type of function represented by the
equation y 4x2 6.
A. absolute value B. linear C. piecewise-defined D
. quadratic
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CCSS
Mathematical Practices 2 Reason abstractly and
quantitatively. 6 Attend to precision.
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Then/Now
You graphed and solved linear equations.

• Solve systems of linear equations graphically.

• Solve systems of linear equations algebraically.

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Vocabulary

• break-even point
• system of equations
• consistent
• inconsistent

• independent
• dependent
• substitution method
• elimination method

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Example 1
Solve by Using a Table

• Solve the system of equations by completing a
  table. x y 32x y 6

Solve for y in each equation.
x y 3 y x 3
2x y 6 y 2x 6
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Example 1
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).
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Example 1
What is the solution of the system of equations?
x y 2x 3y 6
A. (1, 1) B. (0, 2) C. (2, 0) D. (4, 6)
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Example 2
Solve by Graphing

• Solve the system of equations by graphing. x
  2y 0x y 6

Write each equation in slope-intercept form.
The graphs appear to intersect at (4, 2).
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Example 2
Solve by Graphing

• Check Substitute the coordinates into each
  equation.

x 2y 0 x y 6 Original equations
0 0 6 6 Simplify.
Answer The solution of the system is (4, 2).
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Example 2
Which graph shows the solution to the system of
equations below?x 3y 7x y 3
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Example 3
Classify Systems

• A. Graph the system of equations and describe it
  as consistent and independent, consistent and
  dependent, or inconsistent.x y 5x 2y 4

Write each equation in slope-intercept form.
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Example 3
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.
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Example 3
Classify Systems

• B. Graph the system of equations and describe it
  as consistent and independent, consistent and
  dependent, or inconsistent.9x 6y 66x 4y
  4

Write each equation in slope-intercept form.
Since the equations are equivalent, their graphs
are the same line.
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Example 3
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.
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Example 3
Classify Systems

• C. Graph the system of equations and describe it
  as consistent and independent, consistent and
  dependent, or inconsistent.15x 6y 05x 2y
  10

Write each equation in slope-intercept form.
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Example 3
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.
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Example 3
Classify Systems

• D. Graph the system of equations and describe it
  as consistent and independent, consistent and
  dependent, or inconsistent.f(x) 0.5x 2g(x)
  0.5x 2h(x) 0.5x 2

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Example 3
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.
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Concept
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Example 3
A. Graph the system of equations below. What type
of system of equations is shown? x y 52x
y 5
A. consistent and independent B. consistent and
dependent C. consistent D. none of the above
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Example 3
B. Graph the system of equations below. What type
of system of equations is shown? x y 32x
2y 6
A. consistent and independent B. consistent and
dependent C. inconsistent D. none of the above
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Example 3
C. Graph the system of equations below. What type
of system of equations is shown? y 3x 26x
2y 10
A. consistent and independent B. consistent and
dependent C. inconsistent D. none of the above
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Example 3
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.
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Concept
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Example 4
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.
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Example 4
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 The total number of chairs sold
was 48. 265x 320y 13,930 The total amount
earned was 13,930.
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Example 4
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 First equation x 48 y Subtract
y from each side.
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Example 4
Use the Substitution Method

• Solve Substitute 48 y for x in the second
  equation.

265x 320y 13,930 Second equation 265(48
y) 320y 13,930 Substitute 48 y for
x. 12,720 265y 320y 13,930 Distributive
Property 55y 1210 Simplify. y 22 Divide
each side by 55.
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Example 4
Use the Substitution Method

• Now find the value of x. Substitute the value for
  y into either equation.

x y 48 First equation x 22 48 Replace
y with 22. x 26 Subtract 22 from each side.
Answer They sold 26 rocking chairs and 22
Adirondack chairs.
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Example 4
AMUSEMENT PARKS At Amys 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
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Concept
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Example 5
Solve by Using Elimination

• Use the elimination method to solve the system
  of equations.
• x 2y 10x 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.
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Example 5
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).
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Example 5
Use the elimination method to solve the system
of equations. What is the solution to the
system?x 3y 5x 5y 3
A. (2, 1) B. (17, 4) C. (2, 1) D. no solution
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Example 6
No Solution and Infinite Solutions
Solve the system of equations.2x 3y 125x
2y 11

• Read the Test ItemYou are given a system of two
  linear equations and are asked to find the
  solution.

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Example 6
No Solution and Infinite Solutions
Solve the Test ItemMultiply the first equation
by 2 and the second equation by 3. Then add the
equations to eliminate the y variable.
19x 57
x 3
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Example 6
No Solution and Infinite Solutions

• Replace x with 3 and solve for y.

2x 3y 12 First equation 2(3)
3y 12 Replace x with 3. 6 3y 12 Multiply.
3y 6 Subtract 6 from each side. y 2 Divide
each side by 3.
Answer The solution is (3, 2). The correct
answer is D.
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Example 6
Solve the system of equations.x 3y 72x 5y
10
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Concept
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End of the Lesson