Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3.1 Solving Systems Using Tables and Graphs A solution to a system of linear equations is an that makes all of the equations. To solve a system of equations by graphing: 1. each line very carefully. 2. for where the lines intersect. 3. the solution in both equations. Example: Solve each system by graphing 1. y 2x1 1 y x 4 2 2. x4y12 3x4y4 1
Algebra 2 Chapter 3 3.1 Solve Systems Using Tables & Graphs 3. 8x4 y12 2x y 3 4. 7xy6 7x y 6 So what words do we use to describe these systems? Consistent: has at least solution Inconsistent: has solution Independent: has solution Dependent: has solutions. 2
Algebra 2 Chapter 3 3.3 Systems of Inequalities 3.2 Solving Systems Algebraically Sometimes, it is difficult to graph equations accurately by hand. Sometimes, the points of intersection are not integer values and are difficult to see. Sometime, we need a different way to find the solution to a system of equations. A: Solving Systems by Substitution Make a Plan: Example: Solve by substitution This method works well when One equation is already solved for one variable in terms of another like y= or x = 1. x 2y 5 x 2 3y One equation has a term with a coefficient of 1 or -1. Make it Happen: If necessary, solve one equation for one of the variables. Get x = or y = Substitute the expression for the variable in the other equation. 2. 2x y 6 7x 6y 1 Solve the equation. You will almost always have to distribute first. Substitute your solution into either original equation. Solve for the remaining variable. Write your answer as an ordered pair. CHECK YOUR ANSWER IN BOTH EQUATIONS! 3
Algebra 2 Chapter 3 3.3 Systems of Inequalities (Solve by Substitution, Continued) 4x3y10 3. x 2y 10 4. 2x y12 3x7 y1 Make a Plan: This method works well when No variable is already solved for No coefficient is 1 or -1 B: Solving by Elimination 3x2y14 5. 2x2y6 Make it Happen: GOAL to create additive opposites Ex. 4x, -4x or -7y, 7y If necessary, multiply one or both equations by a constant to create additive opposites. Add the two equations. This should eliminate one of the variables. Solve for the variable. Substitute into either original equations and solve for the remaining variable. Write your answer as an ordered pair. CHECK YOUR ANSWER IN BOTH EQUATIONS! 4
Algebra 2 Chapter 3 3.3 Systems of Inequalities (Solve by Elimination) 6. x2y10 3xy9 7. 2x7y4 3x 5y 5 8. x y 2 2x2y0 9. 4xy6 12 x3y18 5
Algebra 2 Chapter 3 3.3 Systems of Inequalities 3.3 Systems of Inequalities A: Graphing a Linear Inequality: (REVIEW) Graph the Inequality like an equation If < or > use a line. If or use a line. Test a point on one side of your boundary line. If the point satisfies the inequality, shade that side of the line. If the point does not, shade the opposite side of the line. In general: If line is y = mx + b < or means shade Graph: y > 3x 4 > or means shade B: Solving a System of Linear Inequalities The solution to a system of equations is the that makes both equations true. The solution to a system of inequalities is the of the two regions that make each inequality true. Examples: What is the solution of the system of inequalities x 3 1. y 4 2. y 2 y 3 x 4 6
Algebra 2 Chapter 3 3.3 Systems of Inequalities (Solve the systems of inequalities) 1 y x3 3. 2 y 2x 4. x 3y 3 x 2y 4 5. y 3 y x4 7
Algebra 2 Chapter 3 3.5 Systems With Three Variables 3.4 Linear Programming A: Terminology Graph the following system of inequalities x 2 3 y 6 x y 10 Some real-world problems involve multiple linear relationships. We call the inequalities the. The overlapping region (the solution to the system) is called the. The corners of the region are called the. In real world problems, you will be trying to MAXIMIZE or MINIMIZE a quantity that depends upon the constraints. Most often this is or. The equation that models what you are trying to maximize or minimize is called the. For the feasible region above, find the maximum and minimum value of the objective function: C = 2x + y 8
Algebra 2 Chapter 3 3.5 Systems With Three Variables B: Linear Programming Procedure Example 1: Find the values of x and y that maximize the objective function for the graph. Example 2: Find the values of x and y that minimize the objective function for the graph. P = 4x y C = x + 9y Example 3: Graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or minimize the objective function. Make it Happen 1. Graph the Inequalities (the constraints) 2. Form the feasible region. 3. Find the coordinates of each vertex. x y 8 2x y 10 x 0 y 0 Maximum for N = 100x + 40y 4. Evaluate the objective function at each vertex. 5. State the Max/Min and where it occurs. 9
Algebra 2 Chapter 3 3.5 Systems With Three Variables Example 4: x2y 6 2x y 7 x 2 y 3 Find the Maximum and Minimum for A = x y 3.5 Systems with Three Variables In lesson 3.1 & 3.2, we learned about systems of 2 equations with 2 variables, that may have a solution (x, y) that makes both equations true. In this lesson, we will learn about systems of 3 equations and 3 variables that may have a solution (x, y, z) that makes all three equations true. 10
Algebra 2 Chapter 3 3.5 Systems With Three Variables Example: Solve each system of equations. Check your answers. Make a Plan GOAL: Use elimination (lesson 3.2) to change your 3 x 3 system to a 2 x 2 system. 1. x y z 1 x y 3z 3 2x y 2z 0 Look for a variable that can easily be eliminated from all three equations. Make it Happen Choose a variable to eliminate. Choose 2 equations & eliminate the variable. Choose a different pair of equations & eliminate the SAME VARIABLE again. You now have TWO new equations each with TWO variables. Use substitution or elimination to solve (Like 3.2) Substitute your solutions into one of the ORIGINAL 3 variable equations and solve for the missing term. Write your answer as an alphabetical ordered triplet. (x, y, z) 11
Algebra 2 2. x y 2 z 7 3 x y 2 z 7 x 3 y z 9 3. x 2 y 3 z 12 2 x y 2 z 5 2 x 2 y z 4 Chapter 3 3.5 Systems With Three Variables 12