Systems of Equations and Inequalities

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Systems of Equations and Inequalities

• Chapter 7

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Aim 7.1 How do we solve systems of linear
equations?

• All equations in the form of Ax By C form a
  straight line when graphed.
• Two such equations are or a linear system.
  systems of equations
• A solution to a system of a linear equations in
  two variables is an ordered pair that satisfies
  both equations.

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Example 1

• Determine if each ordered pair is a solution of
  the following system.
• a. (4, -1) b. (-4, 3)
• X 2Y 2
• X - 2Y 6

• Steps
• Replace the ordered pair in the system for x and
  y.
• Is the equation true?
• If so, then it is a solution.
• If not, then it is not.
• Note It must be true for both equations.

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Solving a System of Linear Equation

• Ways to solve
• By graphing- The point where the lines intersect
  is the solution.
• By Substitution

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Solving by Substitution

• Solve by substitution
• 5x 4y 9
• X 2y -3

• Steps
• 1.Solve either of the equations for one variable
  in terms of the other.
• 2. Substitute the expression from step 1 into the
  other equation.
• 3. Solve the resulting equation.
• 4. Then substitute the vale into one of the
  original equations to solve for the second
  variable.

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Try

• Solve by the substitution method
• 3X 2Y 4
• 2X Y 1

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Solving a System of Linear Equation

• Ways to solve
• By graphing- The point where the lines intersect
  is the solution.
• By Substitution
• By Elimination

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Ex. 2 Solving a System by Addition

• 3x 2y 48
• 9x 8y -24

• Steps
• Rewrite both equations in the form of AX BY
  C.
• If necessary, multiply either equation or both
  equations by appropriate numbers so that the sum
  of the x-coefficients or y-coefficients 0.
• Add the equations
• Solve for one variable.
• Then substitute back into one of the original
  equations and solve for the other variable.

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Guided Practice

• Solve by the elimination method
• 2x 7y 17
• 5y 17 3x

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Analyzing Special Types of Systems

• When lines are parallel there are no points of
  intersection. So the system of linear equations
  has no solution.
• When the equations of the lines are the same then
  you have infinitely many

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Example 3 A System with No Solution

• Solve the system
• 4X 6Y 12
• 6X 9Y 12

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Example 4 Infinitely Many Solutions

• Solve the System.
• Y 3X 2
• 15X 5Y 10

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Applications

• Example 1 A metalworker has some ingots of metal
  alloy that are 20 copper and others that are 60
  copper. How many kilograms of each type of ingot
  should the metalworker combine to create 80 kg of
  a 52 copper alloy?
• Let g mass of the 20 alloy
• m mass of the 60 alloy
• Mass of the alloys g m 80
• Mass of copper 0.2g 0.6m .52(80)
• Now solve for g and m.

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Break- Even Problems

• Suppose a model airplane club publishes a
  newsletter. Expenses are .90 for printing and
  mailing each copy, plus 600 total for research
  writing. The price of the newsletter is 1.50 per
  copy. How many copies of the newsletter must the
  club sell to break even?
• Let x the number of copies
• y the amount of dollars of expenses or
  income
• Expenses are printing costs plus research and
  writing.
• y 0.9x 600
• Income is price times copies sold.
• y 1.5x
• To find out how many copies you need to sell
  solve for x.

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Summary Answer in complete sentences.

• 3- What are three ways to solve systems of
  equations?
• 2- Identify 2 elimination strategies.
• 1-Solve
• Suppose an antique car club publishes a
  newsletter. Expenses are .35 for printing and
  mailing each copy, plus 770 total for research
  and writing. The price of the newsletter is .55
  per copy. How many copies of the newsletter must
  the club sell to break even?

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Aim 7.2 How do we solve systems with three
variables?

• An equation in the form of Ax By Cz D, is
  linear equation with 3 variables.
• Linear variables are x, y and z are the
  variables.
• Example x 2y 3z 9

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Example 1

• Show that the ordered triple (-1, 2, -2) is a
  solution of the system
• X 2y 3z 9
• 2x y 2z -8
• - x 3y 4z 15

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Try

• Show that the ordered triple (-1,-4 , 5) is a
  solution of the system
• X - 2y 3z 22
• 2x 3y - z 5
• 3x y 5z -32

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Example 2 Solving a System in Three Variables

• Solve the system
• 5x 2y 4z 3
• 3x 3y 2z -3
• -2x 5y 3z 3
• Steps
• 1. There are many ways to approach. The central
  idea is to take two equations and eliminate the
  same variable from both pairs.
• 2. Solve the resulting system of two equations in
  2 variables.
• 3. Use back-substitution to find the value of the
  second variable.
• 4. Solve for the third variable.

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Guided Practice

• Solve the system
• X 4Y Z 20
• 3X 2Y Z 8
• 2X 3Y 2Z -16

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Example 3 Solving a System w/a Missing Term

• Steps
• Reduce the system to 2 equations in 2 variables.
• Solve the resulting system of 2 equations in 2
  variables.
• Use back-substitution in 2 variables to find the
  value of the second variable.
• Then find the third variable.

• Solve the system
• X z 8
• X y 2z 17
• X 2y z 16

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Practice

• Solve the system
• 2y z 7
• X 2y z 17
• 2x - 3y 2z -1

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Summary Answer in complete sentences.

• What and how do you solve a system of linear
  equations with 3 variables?
• Give an example from your class work to support
  your explanation.
• Determine if the following statement makes sense,
  and explain your reasoning.
• A system of linear equations in 3 variables, x,
  y, and z cannot contain an equation in the form
• y mx b .

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Aim 7.4 How do we solve systems of nonlinear
equations in 2 variables?

• A system of two nonlinear equations in two
  variables, also called a nonlinear system
  contains at least one equation that cannot be
  expressed in the form Ax By C.
• Example
• X2 2y 10
• 3x y 9

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• A solution of a nonlinear system in two variables
  is an ordered pair of real numbers that satisfies
  both equations in the system.
• The solution set of the system is the set of all
  such ordered pairs.
• Unlike linear systems, the graphs can be circles,
  parabolas or anything other than two lines.
• To solve nonlinear systems we will use the
  substitution method and the addition method.

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Example 1 Solving a Nonlinear System by the
Substitution Method

• Solve
• X2 2Y 10
• 3x Y 9
• Steps 1
• Solve one equation for one variable in terms of
  the other.
• Substitute the expression from Step 1 into the
  other equation.
• Solve the resulting equation containing one
  variable.
• Back substitute the obtained values into the
  equation.
• Check the proposed solution.

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Guided Practice

• Solve by the substitution method
• X2 y -1
• 4x y -1

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Example 2

• Solve by the substitution method
• X Y 3
• (X 2)2 (Y 3)2 4
• (Note This is a circle, with the center at (2,
  -3) and radius 2.)

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Practice

• Solve by the substitution method
• X 2Y 0
• (X 1)2 (Y - 1)2 5

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Example 3 Solving a Nonlinear System using the
Addition Method

• Solve the system
• 4x2 y2 13
• X2 y2 10

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Guided Practice

• Solve the system
• Y X2 5
• X2 Y2 25

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Example 4

• Solve the system
• Y X2 3
• X2 Y2 9

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Guided Practice

• 3x2 2y2 35
• 4x2 3y2 48

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Summary Answer in complete sentences.

• Solve the following systems by the method of your
  choice. Then explain why you chose that method.
• X 3y -5
• X2 Y2 - 25 0
• b. 4X2 XY 30
• X2 3XY -9

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Aim 7.5 How do we solve system of inequalities?

• Graphing a linear Inequality in Two Variables
• Graph 2x 3y gt 6
• Steps
• Replace Inequality with sign and graph the
  linear equation.
• Choose a test point from one of the half planes
  and not from the line. Substitutes its
  coordinates into the inequality.
• Then shade the plane that meets the conditions.

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Practice

• Graph 4x 2y gt 8

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Example 2 Graph the inequality in Two Variables

• Graph

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Example 3

• Graph the following inequalities.
• Hint Graph y 3 and x 2. Graph the region
  that meet the conditions below.
• a. Ylt - 3 (The values less than and equal to 3)
• b. X gt 2( The values greater than 2)

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Example 4 A Nonlinear Inequality

• Graph
• x2 y2 lt 9
• Steps
• Replace inequality with sign and graph.
• Choose a test point from one of the regions not
  on the circle.
• Shade the region that meets the conditions.

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Practice

• Graph x2 y2 gt16

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Example 5Graphing Systems of Linear Inequalities

• Graph the solution set of the system.
• Steps
• Replace each inequality symbol with an equal
  sign. Graph using the x and y-intercepts of each
  equation.
• Note the first equation the line should be ..?
• Whereas the second equations line should
  be?
• 3. Test points in each region to see which
  section is a solution to both inequalities.

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Practice

• Graph the solution set of the system below.

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Example 6 Graphing System of Inequalities

• Graph the solution set of the equation
• Steps
• Graph the first inequality. Note its a solid
  parabola.
• Then graph the second inequality.
• Note the points of intersection of the
  inequalities.
• Now test points and shade the region that makes
  the inequalities true.

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Practice

• Graph the system of inequalities.

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Example 7 Graphing a System of Inequalities

• Graph the solution set of the system

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SummaryAnswer in complete sentences.

• What is a linear inequality in two variables?
• Provide an example with your description.
• How do you determine if an ordered pair is a
  solution of an inequality in two variables x and
  y?
• What is the difference between a dash and a
  solid line in graphing an inequality?
• What is a system of linear inequalities?