Ch. 5 Systems of Equations and Inequalities

Graphing

100

y = 2x + 2

y = x - 1

(-3, -4)

200

y = 2x + 4

y = 3x + 2

(2, 8)

300

x = 3

y = 2x + 1

(3, 7)

400

4x + 2y = 8

2x + y = 4

No solution

500

y = 1/2 x + 2

y = 1/4 x + 4

(8, 6)

Substitution

100

y = 2

3x + 2y = 10

(2, 2)

200

x = 2 -3y

2y = 3 +x

(-1, 1)

300

x - 2y = -10

3x - y = 0

(2, 6)

400

2x – 3y = –2

 4x + y = 24

(5, 4)

500

5x - y = -13

-5x +2y = 24

no solution

Elimination

100

x + 3y = 5

2x -3y = -8

(-1, 2)

200

-5y + 8x = -18

5y + 2x = 58

(4, 10)

300

3y - 5x = -26

-2y - 5x = -16

(4, -2)

400

3x−4y=8

18x−5y=10

(0, -2)

500

4x−9y−2=0

9y = 4x -2

infinitely many solutions

Linear Inequalities

100

What does a dotted line mean when graphing linear inequalities?

> or <

200

How is graphing systems of inequalities different than graphing systems of equations? (2 things)

Sometimes they use dotted lines and you have to shade for inequalities.

300

How do you know where to shade when solving systems of inequalities?

Test a point in the equations. If it's true, shade toward the point. If false, shade away. Where the shaded regions overlap is your solution.

400

y ≤ x−2

y > −3x+5

Shaded toward the bottom right

500

2x – 3y < 12 

x + 5y < 20 

Darkly shaded top left corner

Word Problems

100

The science club charges $4.50 per car at their car wash. Write an inequality to find how many cars they have to wash to earn at least $300.

Use x= number of cars

4.50x ≥ 300

200

Matt and Ming are selling fruit for a school fundraiser. Matt sold 3 small boxes of oranges and 14 large boxes of oranges for a total of $203. Ming sold 11 small boxes of oranges and 11 large boxes of oranges for a total of $220. Write a system of linear equations to model the situation 

S = number of small boxes

L = number of large boxes

3S + 14L = 203

11S + 11L = 220 

300

Ann and Betty together have $60. Ann has $9 more than twice Betty’s amount. Write a system of equations to model the situation 

x= how much Ann has

y= how much Betty has

x + y = 60

x = 2x + 9

400

Tension needs to eat at least an extra 1,000 calories a day to prepare for running a marathon. He has only $25 to spend on the extra food he needs and will spend it on $0.75 donuts which have 360 calories each and $2 energy drinks which have 110 calories.

d= number of donuts

e= number of energy drinks

0.75d + 2e ≤ 25

360d + 110e ≥ 1000

500

Jenny is packing dishes into boxes. Each box can hold either 15 small plates or 8 large plates. She needs to pack at least 9 boxes and at least 150 plates. Write a system of linear inequalities to model the situation

x= boxes with small plates

y= boxes with large plates 

x+y ≥ 9

15x+8y ≥ 150 

*Bonus points* to explain why x ≥ 0 and y ≥ 0 makes sense in the context