Linear Systems Chapter 3 – Algebra 2

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

3.2 Solving Systems Algebraically  Solving Systems of Equations by Substitution  Solve for one of the variables  Substitute it in to find the other variable

 Solving Systems by Elimination  Add the equations together to eliminate one of the variables  May require multiplying one or both equations 3.2 Solving Systems Algebraically

Solving Systems Algebraically

3.2 Solving Systems Algebraically

Warm Up  Maria’s school is selling tickets to a performance. One day they sold 9 senior tickets and 10 student tickets for $215. The next day they sold 3 senior tickets and 5 student tickets for $85. Find the cost for each type of ticket.

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

3-4 Linear Programming EQ: 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:  Graph the constraints to see the solution area  Maximums and minimums occur at the vertices. Test all vertices in the objective function to see which is the max/min.  Vertices are the “Corners” of the solution area. 3-4 Linear Programming EQ: Use Linear Programming to maximize or minimize a function.

 practice: 3-4 Linear Programming EQ: Use Linear Programming to maximize or minimize a function.

 practice:  Homework:  page 138 (7-15)odd  page 144 (1-7) odd 3-4 Linear Programming EQ: 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?

 Suppose you make and sell skin lotion. A quart of regular skin lotion contains 2 c oil and 1 c cocoa butter. A quart of extra-rich skin lotion contains 1 c oil and 2 c cocoa butter. You will make a profit of $10/qt on regular lotion and a profit of $8/qt on extra-rich lotion. You have 24 c oil and 18 c cocoa butter.  a.How many quarts of each type of lotion should you make to maximize your profit?  b.What is the maximum profit?

3-5 Graphs in Three Dimensions EQ: 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 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?  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?  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?  example: Graph 2x + 3y + 4z = 12

Warm Up:  Graph this point in 3D space : (-2, 4, -4)  Show the graph of this line in 3D space:  5x + 6y – 10z = 30 Solve the linear programming system:

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

 x + 4y - 5z = -7  3x + 2y + 3z = 7  2x + y + 5z = 8  Chapter 3 Test on Thursday/Friday  Homework: page 159 (1,5,9,13, 15, 17)