systems of equations & inequalities © beth macdonald 2009

Systems of Equations

& Inequalities

© Beth MacDonald 2009

Systems – Main MENU

• What is a system of equations/inequalities?

• Three types of solutions• Methods used to solve• Answer the question being asked• Dealing with word problems• Practice problems

Pick from above list to learn more.

What is a system?

• Two or more equations (or inequalities) create a system.

• You will be asked to solve at least one system

• We use systems to solve word problems involving multiple items

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Answer the Question Being Asked

• Carefully read how they want your answer– Ordered pair (x, y)– Sum of the solutions

• Add x and y together• Example: x = 3, y = 1, your answer is then 3 + 1 = 4

– Product of the solutions• Multiply x and y together• EX: x = 3, y = 1, your answer is then 3(1) = 3

– Written in a complete sentence• EX: Kathy sold 12 chocolate cakes and 7 vanilla

cakes.

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Three types of SolutionsNo Solution One Solution

(x, y)Infinitely Many

Solutions

Lines never intersect Lines intersect once Lines continuously intersect

Same slope, different y-intercept

Different slopes Same slope, same y-intercept (same line)

y=-x+

3x +

y=3

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Three Methods to Solve

Solve by Graphing

Solve by Substitution

Solve by Elimination

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Solve by Graphing

1. Manually graph both equations on graph paper2. Use the graphing calculator to graph– Graph a system of equations– Graph a system of inequalities

3. If the point of intersection does not have integers for coordinates, find the exact solution by using substitution or elimination.

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Graphing Calculator – pg 1

1. Before you turn on the calculator, solve both equations for y

2. If you’ve never used a graphing calculator, click here to learn about the keys of the calculator

3. Press ON key (bottom left corner) on your graphing calculator

4. Press Y= key (top left corner)5. Type 1st equation into \ Y1=

6. Type 2nd equation into \ Y2=

Keys on the graphing calculator

• Variable x: X,T,θ,n• Negative number: (-) key • x²: x² key• Return to blank screen: 2nd QUIT• erase everything: CLEAR• Delete one character at a time: DEL

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Graphing Calculator – pg 2

7. Press GRAPH (top right corner)can’t see anything… click HERE to change the viewing window

8. Press 2nd CALC (trace key)– Option 5:intersection ENTER– First curve? (move blinker to one of the lines) press

ENTER– Second curve? (blinker should have moved to second

line) press ENTER– Guess? Press ENTER– Intersection (this gives you the x and y coordinates of

the solution)

Graphing Calculator – pg 3

9. How should we answer the question?– Ordered pair solution?– Sum of the solutions?

10. Answer accordingly.

Ready to try some… click HERE

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Viewing WindowOption 1• Press ZOOM• Option #0:ZoomFit (last option listed…use

arrow key to scroll down)• Press ENTER

Option 2• Press WINDOW• Change the Xmin, Xmax, Ymin, Ymax• Press GRAPH

Graphing an Inequality on the calculator

1. Both equations must have y by itself2. Press ON key (bottom left corner) on your graphing calculator3. Press Y= key (top left corner)4. Type 1st equation into \Y1= (the x variable is to the right of the

ALPHA key)

5. Move the cursor to the left of \Y1= so the \ is blinking and press ENTER until the shading is either up (greater than) or down (less than)

6. Type 2nd equation into \ Y2=

7. Follow step 5 for 2nd equation8. Press GRAPH (top right corner)9. Most likely you’ll be asked to find the ordered pair that is

located in the intersection of the two shadings.10. Answer accordingly.

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Substitution

• Use substitution when one of the coefficients is equal to 1.

• You’ll substitute part of one equation into the other equation.

• Once you solve for one variable, you’ll have to use that to solve for the other variable.

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• Line up your x’s, y’s and equal signs• Find a common coefficient for either x or y.• Add or subtract your equations to eliminate

a variable.• Once you solve for a variable, you’ll have

to use that to solve for the other variable.

Elimination

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Word Problems

• AGHHHHH… the dreaded words… don’t be afraid…and don’t skip them!

• Read the question carefully• Underline what sounds important• Try to put yourself into the scenario• Create two equations from the given

information• Pick a method to solve – most likely you’ll use elimination

• Does your answer make sense?

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Practice Problems

• 1• 2• 3• 4• 5• 6• 7• 8• 9Return to

MENU

Practice Problems

1 2 3

4 5 6

7 8 9

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Practice Problem #1

• What is the sum of the solutions fory = x + 33x + y = 5

ANSWER

Answer 1)

• What is the sum of the solutions fory = x + 33x + y = 5

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Practice Problem #2

• Solve. Write your answer as an ordered pair.

-0.5x + y = - 1y - 1 = 2 -7x + 2

ANSWER

Answer 2)Step 1: solve each equation for y

-0.5x + y = - 1 becomes y = 0.5x - 1y - 1 = 2 -7x + 2 becomes y = -7x +5

Step 2: solve by substitution0.5x – 1 = -7x + 57.5x = 6x = 0.8

Step 3: use y - 1 = 2 -7x + 2, substitution 0.8 for x y – 1 = 2 – 7(0.8) + 2 y = 2 – 5.6 + 2 + 1

y = -0.6 Answer: (0.8, -0.6)Return to

Practice Problems

Practice Problem #3

• Solve the system. What is the sum of x and y?

• y = 9x + 20• y = -1/3 x + 13

ANSWER

Answer 3)

• y = 9x + 20• y = -1/3 x + 13

Use substitution9x + 20 = -1/3x +

13+1/3x +1/3x9 1/3x + 20 = 13 Return to

Practice Problems

9 1/3x + 20 = 13 -20 -209 1/3x = -79 1/3 9 1/3 x = -3/4y = 9(-3/4) + 20y = 13.25Sum is -0.75 + 13.25Sum is 12.5

Practice Problem #4

• Solve. What is the product of x and y?

• 2y = 3x + 4• y = -2x - 1.5

ANSWER

Answer 4)

• Solve. What is the product of x and y?

• 2y = 3x + 4• y = -2x - 1.5• Answer: -0.5

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Practice Problem #5

• Clair bought three bars of soap and five sponges for $2.31. Steve bought five bars of soap and three sponges for $3.05. Find the cost of each item.

ANSWER

Answer 5)

Let x = price per bar of soap, y = price per sponge

3x + 5y = 2.315x + 3y = 3.05

x = $0.52y = $0.15

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Practice Problem #6

• Kendra owns a restaurant. She charges $1.50 for 2 eggs and one piece of toast, and $.90 for one egg and one piece of toast. Write and graph a system of equations to determine how much she charges for each egg and each piece of toast. Let x represent the number of eggs and y the number of pieces of toast.

ANSWER

Answer 6)

Let e = price per egg, t = price per slice of toast2e + t = 1.50e + t = 0.90

$0.60 per egg$0.30 for toast

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Practice Problem #7

• Sharon has some one-dollar bills and some five-dollar bills. She has 14 bills. The value of the bills is $30. Solve a system of equations using elimination to find how many of each kind of bill she has.

ANSWER

Answer 7)

Let x = 1 dollar bills, y = 5 dollar bills

x + y = 14x + 5y = 30

4 five-dollar bills10 one-dollar bills

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Practice Problem #8

• Tickets to a local movie were sold at $6.00 for adults and $4.50 for students. There were 240 tickets sold for a total of $1,155.00. Find the number of adult tickets sold and the number of student tickets sold.

ANSWER

Answer 8)

6a + 4.5s = 1155 a + s = 240

50 student and 190 adult tickets

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Practice Problem #9

• Tom has a collection of 27 CDs and Nita has a collection of 18 CDs. Tom is adding 3 CDs a month to his collection while Nita is adding 6 CDs a month to her collection. Write and graph a system to find the number of months after which they will have the same number of CDs. Let x represent the number of months and y the number of CDs.

ANSWER

Answer 9)

Tom: y = 27 + 3xNita: y = 18 + 6x

3 months

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Months

Num

ber

of

CD

’s

10

2

0 3

0 4

0 5

0

1 2 3 4 5 6