Transcript Linear Systems - Western Sierra Collegiate Academy

Linear Systems
Chapter 3 – Algebra 2
3.1 Graphing Systems of Equations
EQ: How do you find the solution to a system by graphing?
3.1 Graphing Systems of Equations
EQ: How do you find the solution to a system by graphing?
Warm Up
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Solve each inequality
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5x – 6 > 24
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-18 – 5y ≥ 52
3.
-5 ( 4x + 1 ) < 23
3-3 Systems of Inequalities
EQ: Show the solution to a system of inequalities
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x – 2y < 6
y ≤ -3/2 x + 5
Steps:
graph each inequality, shading the correct region
the area shaded by both regions is the solution to the
system
3-3 Systems of Inequalities
EQ: Show the solution to a system of inequalities
 Everyone will get a slip of paper with an inequality on it.
 Make sure you know how to graph your inequality.
 Find someone with an equation with a different letter and
draw the solution to your system using colored markers.
Write both of your names and equations on the graph
paper.
 Exchange equations and find a new partner with a
different letter.
 Repeat until you have been part of four graphs!
3-4 Linear Programming
EQ: Use Linear Programming to maximize or minimize a function.
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Linear programming identifies the minimum or maximum
value of some quantity.
This quantity is modeled by an objective function.
Limits on the variable are constraints, written as linear
inequalities.
3-4 Linear Programming
EQ: Use Linear Programming to maximize or minimize a function.
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Example:
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Maximums and minimums occur at the vertices. Test all
vertices in the objective function to see which is the
max/min.
3-4 Linear Programming
EQ: Use Linear Programming to maximize or minimize a function.
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practice:
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Homework:
page 138 (7-15)odd
page 144 (1-9) odd
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Linear Programming
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Cooking Baking a tray of cranberry muffins takes 4 c
milk and 3 c wheat flour. A tray of bran muffins takes 2 c
milk and 3 c wheat flour. A baker has 16 c milk and 15 c
wheat flour. He makes $3 profit per tray of cranberry
muffins and $2 profit per tray of bran muffins.
What is the objective equation?
Write an equation about milk.
Write an equation about wheat.
Graph and solve the system.
How many trays of each type of muffin should the baker
make to maximize his profit?
3-5 Graphs in Three Dimensions
EQ: How do you describe a 3D position in space?
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Adding a third axis – the z axis – allows us to graph in
three dimensional coordinate space.
Coordinates are listed as ordered triples ( x, y, z)
the x unit describes forwards or backwards position
the y unit describes left or right position
the z unit describes up or down position
3-5 Graphs in Three Dimensions
EQ: How do you describe a 3D position in space?
When you graph in
coordinate space, you
show the position of
the point by drawing
arrows to trace each
direction, starting with
x.
3-5 Graphs in Three Dimensions
EQ: How do you describe a 3D position in space?
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Graph each point in coordinate space.
(0, -4, -2)
(-1, 1, 3)
(3, -5, 2)
(3, 3, -3)
3-5 Graphs in Three Dimensions
EQ: How do you describe a 3D position in space?
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The graph of a three variable equation is a plane, and
where it intersects the axes is called a trace.
To graph the trace, you must find the intercept point for
each axis.
To find the x intercept, let y and z be zero.
To find the y intercept let x and z be zero.
To find the z intercept, let x and y be zero.
Plot the three intercepts on their axes, and connect the
points to form a triangle. This triangle is the graph of the
equation.
3-5 Graphs in Three Dimensions
EQ: How do you describe a 3D position in space?
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example: Graph 2x + 3y + 4z = 12
3-6 Solving Systems of Equations in 3 variables
EQ: How do you solve three variable systems?
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To solve a system with 3 variables you need to eliminate
the same variable twice.
Begin by looking at the system and decide which variable
is the easiest to eliminate from ALL three equations.
You will need to eliminate the same variable twice in
order to create a system of two equations in two
variables.
Work backwards to find all three answers
Number the equations to simplify the process.
3-6 Solving Systems of Equations in 3 variables
EQ: How do you solve three variable systems?
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Example:
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x – 3y + 3z = -4
2x + 3y – z = 15
4x – 3y – z = 19
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Which variable is the easiest to eliminate from all three
equations?
3-6 Solving Systems of Equations in 3 variables
EQ: How do you solve three variable systems?
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Solve the system:
2x + y – z = 5
3x – y + 2z = -1
x–y–z=0
Solve the system
2x – y + z = 4
x + 3y – z = 11
4x + y – z = 14
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x + 4y - 5z = -7
3x + 2y + 3z = 7
2x + y + 5z = 8
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Homework: page 159 (1,5,9,13, 15, 17)
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