Chapter 7 Systems and More Graphing

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Chapter 7 Systems and More Graphing

• Section 1 - Systems of Equations and Graphing

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Systems of Equations

• Sometimes its easier to use two different
  variables to represent a situation
• The perimeter of an NBA basketball court is 288
  ft. The length is 44 ft longer than the width.
• Determine the dimensions of the court.

L
L W L W 288
W
2L 2W 288
System of Equations
L W 44

• A system may have
• 1 solution
• No solutions
• Infinitely many solutions

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Solutions to Systems

• A solution to a system has to make all of the
  equations true when substituted.

This ordered pair is a solution to the system.
(L,W)
(94, 50)
2L 2W 288
L W 44
?
94 50 44
94 94
?
188 100 288
288 288
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Solutions to Systems

• A solution to a system has to make all of the
  equations true when substituted.

This ordered pair is not a solution to the system.
(L,W)
(90, 46)
2L 2W 288
L W 44
?
90 46 44
94 94
?
180 92 288
272 ? 288
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Graphing Linear Equations Quick Review
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Graphing Linear Equations - Review
y

• y 3x 2

y 3(2) 2
y 3(1) 2
?

• y 4

• y 1

?
x
1
1
(1, 1)
2
4
(2, 4)
-1
-5
(-1, -5)
?
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Graphing Linear Equations - Review

• Graph using intercepts.
• 3x 2y 12

In every x-intercept, y 0
(4,0)
3x 2(0) 12
?
?
3x 12
x 4
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Graphing Linear Equations - Review

• Graph using intercepts.
• 3x 2y 12

(0,6)
?
?
In every y-intercept, x 0
(4,0)
3(0) 2y 12
?
2y 12
y 6
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Graphing Linear Equations - Review

• Equations like these
• are in slope-intercept format.

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Graphing Linear Equations - Review
4
?
1
slope
?
y-intercept
y-intercept (0,2)
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Solve Systems of Equations

• Graphing Technique

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Find Solution(s) to Systems

• Graphing.

x y 7 y 3x - 1
(2,5)
?
Check
y 3x - 1
x y 7
5 3(2) - 1
2 5 7
5 6 - 1
7 7
5 5
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Find Solution(s) to Systems
y

• Graphing.

x
Parallel lines do not intersect. ?No Solution.
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Find Solution(s) to Systems

• Graphing.

2x 3y 6 8x 12y 24
(0,2)
?
Lines coincide. ?Infinitely many solutions. Any
point on the line would be a solution.
Check
8x 12y 24
2x 3y 6
8(0) 12(2) 24
2(0) 3(2) 6
6 6
24 24
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Review

• Equation Solving Algebraic Method

2x 5 3
5 5
2x 8
2 2
x 4
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Solve Equations with Graphing
Solve this equation graphically.
2x 5 3
2x 5 y
(4,3)
?
3 y
4 x
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Solve Equations with Graphing
Solve this equation graphically.
5 x x 1
5 x y
(3,2)
?
x 1 y
3 x
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Practice

• MathXL Ch 7 Section 1 Homework

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Chapter 7

• Section 2 Systems of Equations and Substitution

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Introduction

• Graphing doesnt always help us get an answer

(?,?)
?
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Substitution Method

• Algebraic Approach

x 7
y
x 3x 1 7
x 3x 1 7
3x 1
y 3x 1
4x 1 7
4x 8
x 2
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Substitution Method

• Algebraic Approach

x 7
y
x 3x 1 7
y 3x 1
y 3(2) 1
4x 1 7
y 6 1
4x 8
y 5
x 2
2
The solution to this system is (2, 5)
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Example

• Solve this system

x 3 2y
y 3x 5
y 3(3 2y) 5
y 9 6y 5
7y 9 5
y 2
7y 14
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Example

• Solve this system

x 3 2y
x 3 2(2)
y 3x 5
x 3 4
x 1
y 2
The solution to this system is (-1, 2)
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Solving for the Variable First

• Solve this system

x 2y 6
2y 2y
3x 2y 4
x 2y 6
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Systems with No Solution

• Solve this system.

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Systems with No Solution

• Solve this system.

This is a false equation. The system has no
solution.
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Systems with Infinite Solutions

• Solve this system.

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Systems with Infinite Solutions

• Solve this system.

Any value of x will make this equation true. The
system has an infinite number of solutions.
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Problem Solving

• The perimeter of a cross-section of a two-by-
  four piece of lumber is 10 ½ inches. The length
  is twice the width.
• Determine the actual dimensions of the
  cross-section of a two-by-four.

2L 2W 10 ½
L 2W
W
L
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Problem Solving
2L 2W 10 ½
2(2W) 2W 10 ½
L 2W
4W 2W 10 ½
L 2(1 ¾ )
6W 10 ½
L 3 ½ inches
W 1 ¾ inches
The cross-sectional width of a two-by-four is 1 ¾
in. and the length is 3 ½ in.
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Practice

• MathXL Ch 7 Section 2 Homework