1. Solving Systems of Equations
Chapter 5

2. Solving Systems of Linear Equations by Graphing
I can solve systems of linear equations by graphing.

3. Solving Systems of Linear Equations by Graphing
Vocabulary (page 134 in Student Journal)
system of linear equations: two or more linear equations with related variables
solution of a system of linear equations: any ordered pair that makes all of the equations in the system true

4. Solving Systems of Linear Equations by Graphing
Examples (space on page 134 in Student Journal)
Determine if the ordered pair is a solution to the system of equations.
(5, 1) b) (-1, -5)
x + y = 6 y = -x + 2
2x – y = 9 y = 3x - 2

5. Solving Systems of Linear Equations by Graphing
Solutions
yes no

6. Solving Systems of Linear Equations by Graphing
Solve the system by graphing. c)

7. Solving Systems of Linear Equations by Graphing
Solution
c) (2, 3)

8. Solving Systems of Linear Equations by Graphing
Example
d) Solve the system of equations by graphing.
y = -2x + 5
y = 4x - 1

9. Solving Systems of Linear Equations by Graphing
Solution
d) (1, 3)

10. Solving Systems of Linear Equations by Graphing
e) A museum sells 4 children’s tickets and 8 adult tickets for their morning show and makes $128. For the afternoon show, they make $72 by selling 6 adult tickets. What is the price of a children’s ticket?

11. Solving Systems of Linear Equations by Graphing
Solution
e) 4x + 8y = 128
6y = 72
x = 8
$8 per child’s ticket

12. Solving Systems of Linear Equations by Substitution
I can solve systems of linear equations by substitution.

13. Solving Systems of Linear Equations by Substitution
Core Concepts (page 139 in Student Journal)
Steps for using Substitution Method
1. solve one equation for a variable
2. substitute the expression into the other equation for the variable and solve the new equation for the variable
3. substitute value of variable into either equation and solve for the other variable

14. Solving Systems of Linear Equations by Substitution
Examples (space on page 139 in Student Journal)
Solve the system by substitution.
y = -x + 3 b) x – y = -4
3y + 5x = x + y = 4

15. Solving Systems of Linear Equations by Substitution
Solutions
(-5, 8)
(0, 4)

16. Solving Systems of Linear Equations by Substitution
c) An theater earns $3640 from a concert. An orchestra ticket costs 3 times as much as a balcony ticket. Below are the sales from a recent concert. What is the price of each type of ticket?

17. Solving Systems of Linear Equations by Substitution
Solution
c) y = 3x
148y + 76x = 3640
x = 21, y = 7
$21 orchestra ticket, $7 balcony ticket

18. Solving Systems of Linear Equations by Elimination
I can solve systems of linear equations by elimination.

19. Solving Systems of Linear Equations by Elimination
Core Concepts (page 144 in Student Journal)
Steps for using Elimination Method
if necessary, multiply one or both equations by a constant
add or subtract to eliminate a variable
3. solve the resulting equations
4. substitute value of variable into either equation and solve for the other variable

20. Solving Systems of Linear Equations by Elimination
Examples (space on page 144 in Student Journal)
Solve the system by elimination.
-2x + 3y = 4 b) 3x + 2y = -2
2x – y = x – 5y = 2

21. Solving Systems of Linear Equations by Elimination
Solutions
(-5, -2)
(-2, 2)

22. Solving Systems of Linear Equations by Elimination
c) A shipping business has 2 locations. Location A has 3 large trucks and 2 small trucks, which cost $270,000. Location B has 4 large trucks and 3 small trucks, which cost $375,000. What is the cost of each type of truck?

23. Solving Systems of Linear Equations by Elimination
Solution
c) 3x + 2y = 270,000
4x + 3y = 375,000
(60000, 45000)
$60,000 for the large truck and $45,000 for the small truck

24. Solving Special Systems of Linear Equations
I can determine the numbers of solutions of linear systems.

25. Solving Special Systems of Linear Equations
Core Concepts (page 149 in Student Journal)
We will get 1 solution when the lines intersect.
We will get no solutions when the lines are parallel.
We get infinitely many solutions when the lines are the same.

26. Solving Special Systems of Linear Equations
Examples (space on page 149 in Student Journal)
Solve the system of linear equations.
y = 3x + 2 b) x – 3y = 6
y = 3x – x – 9y = 18

27. Solving Special Systems of Linear Equations
Solutions
no solution
infinitely many solutions

28. Graphing Linear Inequalities in Two Variables
I can graph linear inequalities in 2 variables.

29. Graphing Linear Inequalities in Two Variables
Vocabulary (page 159 in Student Journal)
linear inequality in two variables: formed by replacing the equal sign in a linear equation with an inequality symbol
solution of a linear inequality: an ordered pair that makes the inequality true

30. Graphing Linear Inequalities in Two Variables
graph of a linear inequality: shows all of the solutions of the inequality in the coordinate plane
half-plane: a region that is bounded by a line.

31. Graphing Linear Inequalities in Two Variables
Core Concepts (page 159 in Student Journal)
Guidelines for Graphing Linear Inequalities
greater than/less than: use dashed line
greater than or equal to/less than or equal to: use solid line
greater than/greater than or equal to: shade above line
less than/less than or equal to: shade below line

32. Graphing Linear Inequalities in Two Variables
Examples (space on page 159 in Student Journal)
Determine whether the ordered pair is a solution to the inequality.
a) 3x – y < 2; (-2, 2) b) 4x – y > 5; (1, 3)

33. Graphing Linear Inequalities in Two Variables
Solutions
yes no

34. Graphing Linear Inequalities in Two Variables
Graph in the coordinate plane.
c) y ≥ -3
d) x + y < 2

35. Graphing Linear Inequalities in Two Variables
Solutions
c) d)

36. Graphing Linear Inequalities in Two Variables
e) You can spend at most $9 for potatoes and carrots for a stew you are making. Potatoes are $3 a pound and carrots are $1.50 per pound. Write and graph an inequality that represents the amount of potatoes and carrots you can buy for the stew.

37. Graphing Linear Inequalities in Two Variables
Solution
e) 3x + 1.5y ≤ 9

38. Systems of Linear Inequalities
I can graph systems of linear inequalities.

39. Systems of Linear Inequalities
Vocabulary (page 164 in Student Journal)
system of linear inequalities: made up of 2 or more linear inequalities
solution to a system of linear inequalities: an ordered pair that makes all the inequalities in the system true

40. Systems of Linear Inequalities
Examples (space on page 164 in Student Journal)
Determine whether each ordered pair is a solution to the system of inequalities.
y > 3x b) y > 3x
y ≤ -x – y ≤ -x – 2
(-2, -1) (0, 4)

41. Systems of Linear Inequalities
Solutions
yes no

42. Systems of Linear Inequalities
Graph the system of inequalities.
c) y ≥ -1 d) x – y > 2
y < -x x – y ≤ - 3

43. Systems of Linear Inequalities
Solutions
c) d)
no solution

44. Systems of Linear Inequalities
Write the system of inequalities from the graph.
e) f)

45. Systems of Linear Inequalities
Solutions
e) y ≤ 2
y > -x + 1
f) x ≥ 1
y ≥ 1/2 x - 2

46. Systems of Linear Inequalities
g) You have at most 7 hours to spend swimming and playing soccer. You want to spend at least 2 hours playing soccer and you want to spend more than 2 hours swimming. Write and graph a system to determine how much time you can spend on each activity.

47. Systems of Linear Inequalities
Solution
g) x + y ≤ 7
x ≥ 2
y > 2