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Linear Systems

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?. 3.2 Solving Systems Algebraically.

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Linear Systems

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  1. Linear Systems Chapter 3 – Algebra 2

  2. 3.1 Graphing Systems of EquationsEQ: How do you find the solution to a system by graphing?

  3. 3.1 Graphing Systems of EquationsEQ: How do you find the solution to a system by graphing?

  4. 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

  5. 3.2 Solving Systems Algebraically • Solving Systems by Elimination • Add the equations together to eliminate one of the variables • May require multiplying one or both equations

  6. Solving Systems Algebraically

  7. 3.2 Solving Systems Algebraically

  8. 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.

  9. 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

  10. 3-3 Systems of InequalitiesEQ: 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!

  11. 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.

  12. 3-4 Linear ProgrammingEQ: Use Linear Programming to maximize or minimize a function. • 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.

  13. 3-4 Linear ProgrammingEQ: Use Linear Programming to maximize or minimize a function. • practice:

  14. 3-4 Linear ProgrammingEQ: Use Linear Programming to maximize or minimize a function. • practice: • Homework: • page 138 (7-15)odd • page 144 (1-7) odd

  15. 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?

  16. 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/qton 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?

  17. 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

  18. 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.

  19. 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)

  20. 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.

  21. 3-5 Graphs in Three DimensionsEQ: How do you describe a 3D position in space? • example: Graph 2x + 3y + 4z = 12

  22. 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:

  23. 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.

  24. 3-6 Solving Systems of Equations in 3 variablesEQ: How do you solve three variable systems? • Example: • x – 3y + 3z = -4 • 2x + 3y – z = 15 • 4x – 3y – z = 19 • Which variable is the easiest to eliminate from all three equations?

  25. 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

  26. 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)

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