1. Systems of Equations and Inequalities
Chapter 6
Systems of Equations and Inequalities

2. Solving systems Graphically
6.1 Linear Systems in Two Variables with Applications
Solving systems Graphically
The solution is (3,1)

3. Solving systems by substitution
6.1 Linear Systems in Two Variables with Applications
Solving systems by substitution
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.

4. 6.1 Linear Systems in Two Variables with Applications

5. 6.1 Linear Systems in Two Variables with Applications
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. 6.1 Linear Systems in Two Variables with Applications

7. Solving Systems Using Elimination
6.1 Linear Systems in Two Variables with Applications
Solving Systems Using Elimination
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.

8. 6.1 Linear Systems in Two Variables with Applications
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

9. 6.1 Linear Systems in Two Variables with Applications

10. 6.1 Linear Systems in Two Variables with Applications
Homework pg

11. Transformations of a Linear System
6.2 Linear Systems in Three Variables with Applications
Transformations of a Linear System
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.

12. 6.2 Linear Systems in Three Variables with Applications
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.

13. 6.2 Linear Systems in Three Variables with Applications
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.

14. 6.2 Linear Systems in Three Variables with Applications
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.

15. 6.2 Linear Systems in Three Variables with Applications
Homework pg

16. Solving a Linear Inequality
6.3 Systems of Inequalities and Linear Programming
Solving a Linear Inequality
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.

17. 6.3 Systems of Inequalities and Linear Programming
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.

18. Solving Linear Programming Applications
6.3 Systems of Inequalities and Linear Programming
Solving Linear Programming Applications
Identify the main objective and the decision variables (descriptive variables may help.)
Write the objective function in terms of these variables.
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.
Graph the constraint inequalities and determine the feasible region.
Identify all corner points of the feasible region and test these points in the objective function to determine the optimal solution(s).

19. S is # of scientific calculators produced
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?
S is # of scientific calculators produced
G is # of graphing calculators produced

20. 6.3 Systems of Inequalities and Linear Programming

21. 6.3 Systems of Inequalities and Linear Programming
(100,170)
(200,170)
(100,100)
(120,80)
(200,80) S

22. 6.3 Systems of Inequalities and Linear Programming
(100,170)
$850
(200,170)
$450
(100,100)
$300
(120,80)
$160
(200,80) $0

23. 6.3 Systems of Inequalities and Linear Programming
Homework pg

24. 6.7 Solving Linear Systems Using Matrix Equations

25. Chapter 6 Review