Linear Systems

Chapter 3 – Algebra 2

3.1 Graphing Systems of EquationsEQ: How do you find the solution to a system by graphing?

3.1 Graphing Systems of EquationsEQ: How do you find the solution to a system by graphing?

Warm Up Solve each inequality

1. 5x – 6 > 24

2. -18 – 5y ≥ 52

3. -5 ( 4x + 1 ) < 23

3-3 Systems of InequalitiesEQ: Show the solution to a system of inequalities

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

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!

3-3 Systems of InequalitiesEQ: Show the solution to a system of inequalities

3-4 Linear ProgrammingEQ: Use Linear Programming to maximize or minimize a function.

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.

Example:

Maximums and minimums occur at the vertices. Test all vertices in the objective function to see which is the max/min.

3-4 Linear ProgrammingEQ: Use Linear Programming to maximize or minimize a function.

practice:

Homework: page 138 (7-15)odd page 144 (1-9) odd

3-4 Linear ProgrammingEQ: Use Linear Programming to maximize or minimize a function.

Linear Programming 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 DimensionsEQ: How do you describe a 3D position in space?

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 DimensionsEQ: 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 DimensionsEQ: How do you describe a 3D position in space?

Graph each point in coordinate space. (0, -4, -2) (-1, 1, 3) (3, -5, 2) (3, 3, -3)

3-5 Graphs in Three DimensionsEQ: How do you describe a 3D position in space?

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 DimensionsEQ: How do you describe a 3D position in space?

example: Graph 2x + 3y + 4z = 12

3-6 Solving Systems of Equations in 3 variablesEQ: How do you solve three variable systems?

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.

Example:

x – 3y + 3z = -4 2x + 3y – z = 15 4x – 3y – z = 19

Which variable is the easiest to eliminate from all three equations?

3-6 Solving Systems of Equations in 3 variablesEQ: How do you solve three variable systems?

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

3-6 Solving Systems of Equations in 3 variablesEQ: How do you solve three variable systems?

x + 4y - 5z = -7 3x + 2y + 3z = 7 2x + y + 5z = 8

Homework: page 159 (1,5,9,13, 15, 17)