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Chapter 6 Systems of Equations and Inequalities 6.1 Linear Systems in Two Variables with Applications Solving systems Graphically 4 x 3 y 9 2 x y 5 The solution is (3,1) 6.1 Linear Systems in Two Variables with Applications Solving systems by substitution 1. Solve one of the equations for x in terms of y or y in terms of x 2. Substitute for the appropriate variable in the other equation 3. Solve the resulting equation for the remaining variable. (This will give you one coordinate of the ordered pair solution.) 4. Substitute the value from step 3 back into either of the original equations to determine the value of the other coordinate. 5. Write the answer as an ordered pair and check. 6.1 Linear Systems in Two Variables with Applications 6.1 Linear Systems in Two Variables with Applications 1. 2. 3. 4. 5. Solve one of the equations for x in terms of y or y in terms of x Substitute for the appropriate variable in the other equation Solve the resulting equation for the remaining variable. (This will give you one coordinate of the ordered pair solution.) Substitute the value from step 3 back into either of the original equations to determine the value of the other coordinate. Write the answer as an ordered pair and check. 6.1 Linear Systems in Two Variables with Applications 6.1 Linear Systems in Two Variables with Applications Solving Systems Using Elimination 1. Be sure each equation is in standard form: Ax+By=C 2. If desired or needed, clear fractions or decimals using a multiplier. 3. Multiply one or both equations by a number that will create coefficients of x (or y) that are additive inverses. 4. Combine the two equations using vertical addition. 5. Solve the resulting equation for the remaining variable. 6. Substitute this x- or y-value back into either of the original equations and solve for the other unknown. 7. Write the answer as an ordered pair and check the solution in both original equations. 6.1 Linear Systems in Two Variables with Applications 1. 2. 3. 4. 5. 6. 7. Be sure each equation is in standard form: Ax+By=C If desired or needed, clear fractions or decimals using a multiplier. Multiply one or both equations by a number that will create coefficients of x (or y) that are additive inverses. Combine the two equations using vertical addition. Solve the resulting equation for the remaining variable. Substitute this x- or y-value back into either of the original equations and solve for the other unknown. Write the answer as an ordered pair and check the solution in both original equations. 6.1 Linear Systems in Two Variables with Applications 6.1 Linear Systems in Two Variables with Applications Homework pg 573 1-74 6.2 Linear Systems in Three Variables with Applications Transformations of a Linear System 1. You can change the order the equations are written. 2. You can multiply both sides of an equation by any nonzero constant. 3. You can add two equations in the system and use the result to replace any other equation in the system. 6.2 Linear Systems in Three Variables with Applications 1. 2 x y 3 z 3 3 x 2 y 4 z 2 4 x 2 y 6 z 7 2. 3. You can change the order the equations are written. You can multiply both sides of an equation by any nonzero constant. You can add two equations in the system and use the result to replace any other equation in the system. 6.2 Linear Systems in Three Variables with Applications 1. x 2 y z 1 x z 3 2 x y z 3 2. 3. You can change the order the equations are written. You can multiply both sides of an equation by any nonzero constant. You can add two equations in the system and use the result to replace any other equation in the system. 6.2 Linear Systems in Three Variables with Applications 1. A 2B 5 B 3C 7 2 A B C 1 2. 3. You can change the order the equations are written. You can multiply both sides of an equation by any nonzero constant. You can add two equations in the system and use the result to replace any other equation in the system. 6.2 Linear Systems in Three Variables with Applications Homework pg 586 1-60 6.3 Systems of Inequalities and Linear Programming Solving a Linear Inequality 1. Graph the boundary line by solving for y and using the slope-intercept form. • Use a solid line if the boundary is included in the solution set. • Use a dashed line if the boundary is excluded from the solution set. 2. For “greater than” inequalities shade the upper half plane. For “less than” inequalities shade the lower half plane. 3. Select a test point from the shaded solution region and substitute the x- and y-values into the original inequality to verify the correct region shaded. 6.3 Systems of Inequalities and Linear Programming 1. 2. 3. Graph the boundary line by solving for y and using the slope-intercept form. • Use a solid line if the boundary is included in the solution set. • Use a dashed line if the boundary is excluded from the solution set. For “greater than” inequalities shade the upper half plane. For “less than” inequalities shade the lower half plane. Select a test point from the shaded solution region and substitute the x- and y-values into the original inequality to verify the correct region shaded. 6.3 Systems of Inequalities and Linear Programming Solving Linear Programming Applications 1. Identify the main objective and the decision variables (descriptive variables may help.) 2. Write the objective function in terms of these variables. 3. Organize all information in a table, with the decision variables and constraints heading up the columns, and their components leading each row. Complete that table using the information given and write the constraint inequalities using the decision variables, constraints, and the domain. 4. Graph the constraint inequalities and determine the feasible region. 5. Identify all corner points of the feasible region and test these points in the objective function to determine the optimal solution(s). 6.3 Systems of Inequalities and Linear Programming A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators much be shipped each day. If each scientific calculator sold results in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize net profits? 2 S 5 G Profit S is # of scientific calculators produced S 100 G 80 G is # of graphing calculators produced S 200 G 170 G S 200 6.3 Systems of Inequalities and Linear Programming S 100 G G 80 S 200 G 170 G S 200 S 6.3 Systems of Inequalities and Linear Programming S 100 G 80 S 200 G (100,170) (200,170) G 170 G S 200 (100,100) (120,80) (200,80) S 6.3 Systems of Inequalities and Linear Programming 2 S 5 G Profit (100,170) $850 (200,170) $450 (100,100) (120,80) $300 $160 (200,80) $0 6.3 Systems of Inequalities and Linear Programming Homework pg 600 1-60 6.7 Solving Linear Systems Using Matrix Equations Chapter 6 Review