cc math i unit 7 systems of equations and...

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CC Math I UNIT 7 Systems of Equations and Inequalities

Name_____________________________ Teacher _______________________ Estimated Test Date______________

MAIN CONCEPTS Page(s)

Study Guide 1 – 2

Equations of Circles & Midpoint 3 – 5 Parallel and Perpendicular Lines 6 – 8

Systems of Equations 9 – 12

Applications of Linear Systems 13 Systems of Linear Inequalities 14 – 16

Unit 11 Review 17 – 21 Selected Answers 22 – 28

Essential Questions:

How do you determine when more than one equation or inequality is needed to solve a problem?

What are the different methods of solving systems of equations or inequalities?

How do you determine the best method to apply?

What are the different possible solution sets of a system of linear equations or inequalities?

What is the significance of the solution for a system of equations or inequalities?

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Common Core Standards

8.EE.8 Analyze and solve pairs of simultaneous linear equations.

a. Understand that solutions to a system of two linear equations in two variables correspond to points of

intersection of their graphs, because points of intersection satisfy both equations simultaneously.

b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by

graphing the equations. Solve simple cases by inspection.

c. Solve real-world and mathematical problems leading to two linear equations in two variables.

A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations

on coordinate axes with labels and scales.

A-CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and

interpret solutions as viable or non- viable options in a modeling context.

A-REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y =

g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using

technology to graph the functions, make tables of values, or find successive approximations. Include

cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic

functions.

A-REI.12 Graph the solutions to a linear inequality in two variables as a half- plane (excluding the boundary in the

case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as

the intersection of the corresponding half-planes.

A-REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that

equation and a multiple of the other produces a system with the same solutions.

A-REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of

linear equations in two variables.

G-CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the

undefined notions of point, line, distance along a line, and distance around a circular arc. Note: At this

level, distance around a circular arc is not addressed.

G-GPE.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a

figure defined by four given points in the coordinate plane is a rectangle.

G-GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems

(e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given

point).

G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given

ratio.

Students will understand that…

the equation of a circle can be derived from the Pythagorean Theorem

parallel lines have the same slope and perpendicular lines have opposite reciprocal slopes

the midpoint can be calculated and applied to find an endpoint

two equations can have one, none, or infinitely many solutions

when solving a system of equations by graphing the solution is the intersection point

substitution of an algebraic expression can be used when you have the X or Y variable isolated

linear equations in standard form can be added or subtracted to combine or eliminate a variable by making zero

one or both equations in a system of equations can be multiplied in order to get the coefficients of X or Y to be the

same or inverses

real world problems involving two variables can be solved using systems of equations

linear inequalities can be graphing similar to linear equations, but shading and line determination needs to be

accounted for

two or more linear inequalities can be graphed on the same coordinate plane and their solutions are the

overlapping sections

real world problems involving two variables and inequalities can be modeled using a system of linear inequalities

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Equations of Circles

1. What is the equation for any circle centered on the origin with a radius of r units?

2. How can this be proven by the Pythagorean Theorem?

3. Why do the slopes of perpendicular diagonal lines have a product of –1?

4. Are circles functions? Explain.

Think about a point (x,y) starting at (5,0) and moving

counterclockwise, tracing around the circle.

5. How does the y-coordinate of the point change as the x-

coordinate approaches zero and then becomes negative?

6. Draw a point at (3,4) and create a right triangle with that

point and (0,0). Draw a point at (–4,3) and create a right

triangle with that point and (0,0). How do the triangles change

as the point moves around the circle?

7. Use your knowledge of right triangles and circles to write an

equation relating x and y to the radius, 5.

8. How would you change your equation from #7 if the radius of the circle was 3 units?

9. Inscribe a rectangle in the circle at the following coordinates (5, 0), (4,3), (–5, 0), and (–4, –3). What are

the slopes of the two shorter sides?

10. What are the slopes of the two longer sides?

11. What is the product of the slopes of two perpendicular sides?

12. If you pick any point on the circle (x,y) in which x≠0 and y≠0, and connect it to (r,0) and (–r,0) where r

is the radius, what kind of triangle do you make?

13. Using the variables and the triangle in #12, calculate the slope of each of the two perpendicular sides.

Let (x,y) be (x2, y2) and let the point with the x-intercept be (x1, y1). m = (y2 – y1) / (x2 – x1)

14. Calculate the product of the two slopes.

15. What is the equation of the circle centered at the origin and has a radius of r units?

16. Solve the equation from #15 for y2.

17. Substitute the value you calculated for #16 into the product you calculated in #14. Simplify the product.

x

y

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Circles and Midpoint

The midpoint of a line segment is the point in the middle. It is equidistant from each of the two endpoints.

It can be calculated by averaging the endpoints’ x-values and y-values.

For example, a line segment with endpoints at (–2, 7) and (4, –11) would have a midpoint at

( (–2+4)/2 , (7 – 11) /2) = (1, –2). The midpoint’s x-value is equidistant from each endpoint’s x-value, and

the midpoint’s y-value is equidistant from each endpoint’s y-value.

Each of the sets for 1 – 4 represents the endpoints of a circle’s diameter. For each set of endpoints,

1) Calculate the midpoint. Clearly show your work.

2) Calculate the diameter. Clearly show your work.

3) Calculate the circumference of the circle. Clearly show your work.

4) Calculate the area of the circle. Clearly show your work.

5) Write the equation of the circle if the center was on the origin.

Bonus: calculate the equation of the circle with the diameter on the given endpoints. Use what you’ve

learned about the equation of a circle centered at the origin as well as what you’ve learned about

translations of functions.

1. {(–8, 11), (4, –5)}

Midpoint:

Diameter:

Circumference:

Area:

Equation if the circle was centered on the origin:

Bonus: What is the equation of the given circle?

2. {(4, 7), (–2, –1)}

Midpoint:

Diameter:

Circumference:

Area:



3. {(–9, –14), (1, 10)}

Midpoint:

Diameter:

Circumference:

Area:



4. {(–1, 8), (23, 26)}

Midpoint:

Diameter:

Circumference:

Area:



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Midpoint Practice: 1 – 20: Calculate the midpoint of the line segment with the given

endpoints. 21 – 26: Calculate the other endpoint of the given line segment.

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Parallel and Perpendicular Lines

In an earlier lesson, you used a circle to show that parallel lines have the same slope, but don’t share any

points. You also learned that perpendicular diagonal lines have slopes that are opposite reciprocals of one

another and a product of –1. Use that knowledge to answer the following.

Are the two lines parallel, perpendicular, or neither?

1. y = 3 2. y = 2x – 1 3. x = 1

y = 2 6x – 3y = 3 y = 2

4. x = 4 5. 𝑦 =1

5𝑥 − 1 6. y = 3x

x = 8 𝑦 = −5𝑥 − 1 y = 3x + 1

Write the equation of the line described in point-slope form.

7. the line parallel to y = 4x – 3 and passes through (2, 7)

8. the line parallel to 3x – 4y = 4 and passes through (1, 2)

9. the line parallel to 4x + 2y = 6 and passes through (1, 5)

10. the line parallel to x = 3 and passes through (1, 7)

11. the line perpendicular to x + 2y = 2 and passes through (2, 3)

12. the line perpendicular to y = 3x – 4 and passes through (1, 1)

13. the line perpendicular to y = 6 and passes through (5, 2)

14. the line perpendicular to 𝑦 = −2

5𝑥 −

7

9 and passes through (2, 8)

15. A line segment has endpoints at (–5, 8) and (7, –2). What is the equation of the line that is

perpendicular to the line segment and passes through its midpoint? (perpendicular bisector)

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Solving Systems of Linear Equations by Graphing

A system of equations is a group of two or more equations that use the same set of variables. The solutions

to a system of equations are the values for each variable that satisfy each equation in the system. Systems

of equations are also referred to as simultaneous equations. There are also systems of inequalities. Systems

can have anywhere between zero and infinite solutions.

There are several methods for solving systems of equations, but the three that you should master are the

following: graphing, substitution, and elimination (combination). Know how to solve any system by any of

the three methods. Also know which method is the most appropriate to use for various systems of

equations.

Arithmetic errors and sign errors are the most common errors made by students in the past. Carefully check

your calculations by using inverse operations. Carefully watch the sign of constants and coefficients. It is a

good idea to use the definition of subtraction to rewrite all subtraction as addition.

1. A rectangle’s length is one inch shorter than twice its width. Its perimeter is 10 inches. What is

the rectangle’s area? Let L = the length of the rectangle in inches. Let W = the width of the

rectangle in inches.

Rewrite the word problem as a system of equations.

Solve each system of linear equations by graphing. To solve a system of equations by graphing, graph each

equation and find the point(s) of intersection. Write your solution as an ordered pair.

2. y = 2x – 3 4. y = –x – 2

y = 22

3x y = – x + 2

3. x – y = 3 5. y = – 2x + 1

x – 2y = 1 4x + 2y = 2

𝐵𝑜𝑛𝑢𝑠. 𝐴 = 1 −1

𝐵 ; 𝐵 = 1 −

1

𝐶 𝑤ℎ𝑒𝑟𝑒 𝐴, 𝐵, 𝑎𝑛𝑑 𝐶 𝑎𝑟𝑒 𝑛𝑜𝑛 − 𝑧𝑒𝑟𝑜 𝑛𝑢𝑚𝑏𝑒𝑟𝑠.

𝐶𝑎𝑙𝑐𝑢𝑙𝑎𝑡𝑒 𝑡ℎ𝑒 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝐴𝐵𝐶. 𝐶𝑙𝑒𝑎𝑟𝑙𝑦 𝑠ℎ𝑜𝑤 𝑜𝑟 𝑒𝑥𝑝𝑙𝑎𝑖𝑛 𝑦𝑜𝑢𝑟 𝑤𝑜𝑟𝑘.

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Solving Systems of Linear Equations by Substitution Substitution works best when at least one of the equations is already solved for a variable.

Steps for solving systems of linear equations by using substitution.

Step 1: If none of the equations in the system are already solved for a variable, solve one of the equations

for a variable (or variable expression) that can be easily substituted into the other equation.

Step 2: Use the substitution property of equality to substitute the value of the variable from the first

equation into the second equation. As always, use grouping symbols when you substitute.

Step 3: Simplify each side of the equation. Carefully use the distributive property and carefully watch your

signs. If one of the factors or terms is negative, it might be safer to rewrite the subtraction problem as an

addition problem. Substitute the value for the variable you calculated back into one of the original

equations. Solve for the other variable.

Step 4: Check your answer with the original system. Write your final answer.

Solve each system by using substitution. Remember to check your answers before using the answer key.

Write your final answer as an ordered pair.

1. 2. 3.

4. 5. 6.

Solve the first equation for y. Substitute it into the second equation.

7. 8.

Graph the system to estimate the solution. Use substitution to find the exact solution.

9.

Solving Systems of Linear Equations by Elimination (Combination)

y = 2x – 5

y = 3x – 2

𝑦 =1

2𝑥 + 1

1

4

6x – 2y = 5

𝑦 = −1

3𝑥 − 2

2x – 6y = 12

𝑦 =4

11𝑥 + 10

4x + 5y = 2

𝑦 =1

4𝑥 + 3

x – 4y = 12

f(x) = 5x – 7

f(x) = x – 1

3x – 2y = 2

x – 3y = 4

2x + 5y = 10

4x 5y = 2

y = 0.4x – 1

y = 1.2x + 1

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Another way to solving systems of equations is to eliminate variables until only one variable is left. This

can be done by using the addition or subtraction property of equality.

Steps for solving systems by elimination / combination:

Step 1: Write the equations in the same form. Normally standard form is used for elimination.

Step 2: Use the multiplication or division properties of equality to make the coefficients of one variable

either opposites or the same. Sometimes you will have to multiply or divide each side of both equations in

the system in order to get opposite or same coefficients for a single varaible. If the side has multiple terms,

correctly show the distributive property.

Step 3: Combine the two equations into one by using the addition or subtraction property of equality.

Carefully watch your signs. Be extra careful if you are using subtraction along with the distributive

property. Substitute the value for the variable you calculated back into one of the original equations. Solve

for the other variable.

Step 4: Check your answer with the original system. Write your final answer. For systems like ones below,

write your final answer as an ordered pair.

Solve by using elimination. Remember to check your answers before using the answer key.

1. 2. 3.

4. 5. 6.

7. 8. 9.

3x + 2y = 3

3x – 4y = 9

4x – 3y = 7

2x + 3y = 5

3x + 5y = 6

3x y = 6

5x + 2y = 7

x + 4y = 4

9x – 18y = 27

7x + 2y = 1

5x 3y = 3

5y + 7x = 4

7x + 10y = 8

8x + 40y = 48

9x + 2y = 11

6x + 3y = 42

8x + 15y = 2

2x – 3y = 13

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33. Systems of linear equations can have one of three solutions: 1 solution, no solution, or

infinitely many solutions. Explain how each of the three kinds of solutions are possible.

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Declare variables, set up a system of equations, and answer the question.

1. Al shopped at the clothing store next to Willy’s Fun Arcade on Monday. That week the store was having

a sale where all shirts were on sale for the same price and all pairs of pants were on sale for the same price.

Al bought four shirts and three pairs of pants. His bill before taxes was $85.50. Al returned on Tuesday and

bought three more shirts and five more pairs of pants. His bill before taxes was $115. What was the price of

each shirt and each pair of pants?

2. The Box Emporium sells small boxes for $1.10 each and large boxes for $2.35 each. One day, the

receipts showed that 172 boxes were sold for a total of $294.20. How many large boxes were sold?

3. A test was worth 100 points and had 40 questions. Some questions were worth 2 points each and others

were worth 4 points each. How many questions on the test were worth 2 points?

4. Harvey spent $31.50 on supplies to start his lemonade stand. To make a pitcher of lemonade, he uses

$0.40 worth of materials. He sells each pitcher of lemonade for $2.50. How many pitchers of lemonade

must he sell to break even?

5. The math club and the science club each had fundraisers to buy supplies for a local hospice. They each

bought cases of bottled water and cases of juice from the same store and they paid the same price. The math

club spent $135 (before taxes) to buy six cases of juice and one case of bottled water. The science club

spent $110 (before taxes) to buy two cases of bottled water and four cases of juice. How much did one case

of bottled water cost? How much did one case of juice cost?

6. Rocky spends 250 minutes each day studying. His ratio of time spent on math to time spent on all other

classes is 3 to 2. How many minutes each day does he spend on math?

7. A bowl of Skittles has 660 delicious pieces of candy, but only grape Skittles and lime Skittles are left.

The ratio of grape Skittles to Lime skittles is 11 to 4. How many of the Skittles are lime?

8. Andy paddles his canoe 12 miles downstream (with the current) in 2 hours. He then turns his canoe

around and paddles 12 miles upstream (against the current) in 3 hours. Assuming Andy paddles at a

constant rate and the stream is flowing at a constant rate, how fast is he paddling?

9. A chemistry student needs 30 ml of a 12% salt solution. He has two salt solutions, A and B, to mix

together to form the 30 ml solution. Salt solution A is 11% salt and salt solution B is 15% salt. How much

of each solution should be used?

10. There are 1170 students in a school. The ratio of girls to boys is 23:22. How many more girls are there

than boys?

11. Calvin has $8.80 in pennies and nickels. If there are twice as many nickels as pennies, how many

pennies does Calvin have? How many nickels?

12. The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased

by 27. Find the number.

13. Two small pitchers and one large pitcher can hold 8 cups of water. One large pitcher minus one small

pitcher constitutes 2 cups of water. How many cups of water can each pitcher hold?

14. Adam is 20 years younger than Brian. In two years Brian will be twice as old as Adam. How old are

they now?

15. Carmen is 12 years older than David. Five years ago the sum of their ages was 28. How old are they

now?

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Linear Inequalities

You have spent a lot of time dealing with linear functions. Function rules are equations. However

sometimes real life situations can only be modeled by inequalities. If a line in the form of y=mx + b is

where equality is true, then all of the points on one side of the line represent y < mx + b, and all of the

points on the other side of the line represent y > mx + b.

Graphing a system of linear inequalities

Follow the steps for each inequality in the system.

1) Solve the inequality for y. If there is no range variable, solve for the domain variable. Be careful

when using the properties of inequality and the distributive property. Carefully watch your signs.

2) Use your knowledge of linear equations to calculate 2 points on the boundary line of the

inequality.

3) CHECK before you graph. Substitute the two points from the boundary line back into the original

inequality. Each of the two points should make the original inequality a true equation.

4) If the points on the boundary line are part of the solution set (<, >) the boundary line should be

solid. (All points graphed are solutions.) If they are not part of the solution set (<, >) the boundary

line should be dashed. Graph the correct boundary line.

5) Identify which side of the line represents “greater than” and which side represents “less than”.

Use the y-axis to do this quickly.

6) CHECK: Substitute any point from the correct region back into the original inequality. It should

make the inequality true.

7) Shade the correct region. All of these points are solutions of the inequality.

Once you do complete these steps for each inequality, make sure that the overlapping region (the solution

to the system) is clearly marked.

Identifying a system of inequalities from a graph.

Follow these steps for each of the graphed inequalities.

1) Identify two points on the boundary line. Use the points to calculate the slope of the boundary line.

2) Use the slope “m” and one of the points (x,y) to calculate the equation of the boundary line.

3) Identify the inequality symbol by checking whether the line is solid or dashed and which side is

shaded.

4) CHECK: The two points from the boundary line should make each side of the inequality equal. A

point from the shaded region should make the inequality true.

5) If you need to write the inequality in standard from (solve for the constant), use the properties of

inequality to isolate the constant term on one side of the inequality. Use addition and the

commutative property of addition to get the x term before the y term. If there are any fractions,

multiply each side by the LCD. Carefully watch your signs, and remember to reverse the

inequality symbol if you multiply or divide both sides of the inequality by a negative quantity. Do

step 4 again with the system written in standard form.

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Write a system of linear inequalities for each of the following.

1. Quadrant II 2. Quadrant IV 3. Quadrant I 4. Quadrant III

Graph the system of linear inequalities. Do a full check.

5. 6x + 4y < 8 6. 3x – 2y > 4

y ≥ 2x + 1 y < x + 1

Declare variables and write a system of linear inequalities. Graph and check.

7. Mr. Wilson is going to bake two apple pies. He needs at least two pounds of apples for each pie. The

only apples on sale at the grocery store are Gala apples for $1.00/pound and Granny Smith apples for

$1.50/pound. Mr. Wilson wants to spend less than $6.00 on the apples. Let x = pounds of Gala apples. Let

y = pounds of Granny Smith apples. How many pounds of each type of apple can he purchase?

Are all of the points in the shaded region solutions to the problem?

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8. Sally has seven seashells. Some are supersized and some are small. She sells the supersized seashells for

$2 each and the small seashells for $0.50 each. Let x = # of small seashells. Let y = # of supersized

seashells. She won’t leave the seashore until she earns at least $9, so how many small and supersized

seashells does she need to sell?


Write a system of linear inequalities for each graph.

9. 10.

11. 12.

y

x

y

x

y

x

y

x

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Unit 7 Practice Assessment 1. Calculate the midpoint of the line segment with endpoints at (–8, 3) and (14, –9)

_________________ Clearly show your work.

2.Graph the given system of inequalities

𝑦 ≥ −1

2𝑥 + 2

y < 3x – 1

Fill in each blank (solid or dashed) to make each statement true.

𝑦 ≥ −1

2𝑥 + 2 has a ___________ boundary line.

y < 3x – 1 has a ______________ boundary line.

3. Solve the system of equations by graphing.

𝑦 = −1

3𝑥 + 6

𝑦 = 2𝑥 − 1

4 – 7: Solve each system by using substitution or elimination. Pick the method that is most

appropriate for each system.

4. y = 2x + 10

y = 3x + 13

5. x + 3y = –5

4x – 3y = 10

6. y = x + 5

x + 3y = 11

7. x + 3y = –10

x + y = –4

8. Write the equation of a circle with its center at (0,0) and a radius of 6 units.

Write the equation of a circle with its center at (0,0) and a diameter of 20 units.

9. What is the slope of the line that is parallel to y = 4x + 2?

What is the equation of the line that passes through (2, 3) is parallel to y = 4x + 2?

10. What is the slope of the line that is perpendicular to 𝑦 =3

5𝑥 + 7

What is the equation of the line that passes through (6, 5) is perpendicular to 𝑦 =3

5𝑥 + 7

11. Solve the following systems:

a.

823

534

yx

yx

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b.

1536

52

yx

xy

c.

4

2

yx

yx

12. Each of the graphs below represent the graphing of a system of two equations. How many

solutions are represented in graphs a, b, and c?

Graph a: Graph b: Graph c:

Graph the following inequalities.

Rewrite these equations in slope-intercept form or use x and y

intercepts.

13.

x 2

y x - 4

14.

– y 3x + 4

-3x + 3y -9

a. b. c.

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15. The concession stand is selling hot dogs and hamburgers during a game. At halftime, they sold a

total of 78 hot dogs and hamburgers and brought in $105.50. How many of each item did they sell

if hamburgers sold for $1.50 and hot dogs sold for $1.25?

Hamburgers:______________ Hot Dogs: ______________

16. There are two different jobs Jordan is considering. The first job will pay her $4200 per month

plus an annual bonus of $4500. The second job pays $3700 per month plus an annual bonus of

$500. How many months of employment would both jobs be the same salary?

17. The length of a rectangle is 4.3 cm more than 5 times the width. If the perimeter of the

rectangle is 80.6 cm, what are its dimensions? Show all work!

Length: _______________ Width: __________

18. A garden supply store sells two types of lawn mowers. Total sales of mowers for the year were

$8779.70. The total number of mowers sold was 30. The small mower costs $249.99. The large

mower costs $329.99.

a. Use x to represent the number of small mowers sold and y to represent the number of

large mowers sold. Write a system of equations that represent the number of mowers

sold and the total sales for the year.

b. Use your equations to find the number sold of each type of mower.

19. Marcella and Rupert bought some party supplies. Marcella bought 3 packages of balloons and 4

packages of favors for $14.63. Rupert bought 2 packages of balloons and 5 packages of favors for

$16.03. Find the price of a package of balloons and a package of favors.

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20. Which system describes the following situation? Craig has $0.80 in nickels n and dimes d. He has

four more nickels than dimes.

a. d + n = 4 b. n – d = 4

10d + 5n = 80 10d + 5n = 80

c. d – n =4 d. d + n = 4

10d + 5n = 80 10d – 5n = 80

Graph the linear inequality.

21. y < -x + 2 22. y ≥ 2

3 x – 3

23. Is (2, 2) a solution to the linear inequality y ≤ –2x + 1?

Graph the systems of linear inequalities.

24. y > 1

4x 25. y > 2x + 4

y ≤ –x + 4 2x – y ≤ 4

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26. Is (1, 19) a solution the system of equations?

𝑦 ≤ 7𝑥 − 13

𝑦 > 3𝑥 + 6

27. Suppose you buy flour and cornmeal in bulk to make flour tortillas and corn tortillas. Flour costs

$1.50/lb. Cornmeal costs $2.50/lb. You want to spend less than $9.50 on flour and cornmeal, and you

need at least 4 lb altogether.

a. Write a system of two linear inequalities that describe the situation.

b. Graph the system to show all possible solutions.

c. Write two possible solutions.

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Answers: Pages 6 (1 – 14)

1. y = 3 2. y = 2x – 1 3. x = 1

y = 2 6x – 3y = 3 y = 2

The 2 horizontal lines are parallel. 6𝑥 + (−6𝑥) − 3𝑦 = −6𝑥 + 3 The vertical line is

−3𝑦

−3=

−6𝑥+3

−3 perpendicular to the horizontal line.

y = 2x – 1

neither, they are the same line

4. x = 4 5. 𝑦 =1

5𝑥 − 1 6. y = 3x

x = 8 𝑦 = −5𝑥 − 1 y = 3x + 1

The 2 vertical lines are parallel. perpendicular, the product parallel, the slopes

of the slopes is 1; (1

5)(−5) = −1 are the same (3=3)

and the y-intercepts are

different (1≠0)

7. the line parallel to y = 4x – 3 and passes through (2, 7) ; y – 7 = 4(x + 2)

Check the y-intercepts; y – 7 = 4 [(0) + 2]; y – 7 = 4(2) ; y – 7 + 7 = 8 + 7; y = 15; 15 ≠ 3

8. the line parallel to 3x – 4y = 4 and passes through (1, 2) ; 3𝑥 + (3𝑥) − 4𝑦 = 3𝑥 − 4; −4𝑦 = −3𝑥 − 4 − 4𝑦

−4=3𝑥 − 4

−4; 𝑦 =

3

4𝑥 + 1

The parallel line: 𝑦 + 2 =3

4(𝑥 − 1)

Check the y-intercepts; 3(0) 4y = 4; 4y = 4; − 4𝑦

−4=

−4

−4 ; y = 1

𝑦 + 2 =3

4[(0) − 1]; 𝑦 + 2 =

3

4(1); 𝑦 + 2 − 2 = −

3

4− 2; 𝑦 = −2

3

4 ; −2

3

4≠ 1

9. the line parallel to 4x + 2y = 6 and passes through (1, 5) ; 4𝑥 + (−4𝑥) + 2𝑦 = −4𝑥 + 6; 2𝑦 = −4𝑥 + 6 2𝑦

2=

−4𝑥+6

2; 𝑦 = −2𝑥 + 3

The parallel line: y – 5 = 2(x + 1)

Check the y-intercepts; y – 5 = 2[(0) + 1]; y – 5 = 2(1) ; y – 5 = 2 ; y – 5 + 5 = 2 + 5 ; y = 3

3 = 3; The two lines are not parallel. They are the same line.

10. the line parallel to x = 3 and passes through (1, 7)

Vertical lines are parallel. 3 ≠ 1; x = 1

11. the line perpendicular to x + 2y = 2 and passes through (2, 3) ; 𝑥 + (−𝑥) + 2𝑦 = −𝑥 + 2; 2𝑦 = −𝑥 + 2 2𝑦

2=

−𝑥 + 2

2 ; 𝑦 = −

1

2𝑥 + 1 ; (−

1

2) (2) = −1

y + 3 = 2(x + 2)

12. the line perpendicular to y = 3x – 4 and passes through (1, 1) ; 3 (−1

3) = −1

𝑦 + 1 = −1

3(𝑥 − 1)

13. the line perpendicular to y = 6 and passes through (5, 2)

Horizontal lines are perpendicular to vertical lines. x = 5

14. the line perpendicular to 𝑦 = −2

5𝑥 −

7

9 and passes through (2, 8) ; (−

2

5) (

5

2) = −1

y – 8 = 5

2(x 2)

23 | P a g e

Answers Page 10

Remember to check your answers before using the answer key.

1. 2. 3.

4. 5. 6.

3x – 2 = 2x – 5

+2 +2

3x – 2x = 2x – 2x – 3

x = 3; y = 2(3) – 5

y = 6 – 5 ; y = 11

(3, 11)

6x – 2(1

2𝑥 +

5

4) = 5

6x – x 5

2 = 5

5𝑥 −5

2+

5

2= 5 +

5

2

(1

5) (5𝑥) = (

15

2) (

1

5)

𝑥 = 11

2; 6 (

3

2) – 2y = 5;

9 – 2y = 5; 9 – 9 – 2y = 5 – 9

2y = 4

(−1

2) (−2𝑦) = (−4) (

−1

2); y = 2

(11

2, 2)

I will show each step of the

distributive property and the

definition of subtraction.

2x – 6(−1

3𝑥 − 2) = 12

2x + (6) [−1

3𝑥 + (−2)] = 12

2x + (6)(−1

3𝑥)+ (6)(2)=12

2x + 2x + 12 = 12

12 = 12; Identity

Infinitely Many Solutions.

4x + 5(4

11𝑥 + 10) = 2

4𝑥 +20

11𝑥 + 50 = 2

4𝑥 +20

11𝑥 + 50 = 2

64

11𝑥 + 50 − 50 = 2 − 50

(11

64) (

64

11𝑥) = (−48) (

11

64)

𝑥 = −33

4= −8

1

4

𝑦 = (4

11) (−

33

4) + 10

𝑦 = −3 + 10 ; 𝑦 = 7

(−81

4 , 7)

𝑦 =1

4𝑥 + 3

x – 4( 1

4𝑥 + 3) = 12

x – x – 12 = 12; 12 ≠ 12

no solution

5x – 7 = x – 1

5x – x – 7 = x – x – 1

4x – 7 = 1; 4x – 7 + 7 = 1 + 7

4x = 6; 4𝑥

4=

6

4; 𝑥 = 1

1

2

𝑓 (11

2) = (1

1

2) − 1; 𝑓 (1

1

2) =

1

2

(11

2 ,

1

2)

24 | P a g e

Answers: Page 10

Graph the system to estimate the solution. Use substitution to find the exact solution.

9.

x is between 1 and 2. y is between 0 and –1.

0.4x – 1 = 1.2x + 1

0.4x + 1.2x – 1 = 1.2x + 1.2x + 1

1.6x – 1 + 1 = 1+1; 1.6x = 2

y = 0.4x – 1

y = 1.2x + 1

7. 3𝑥 − 2𝑦 = −2 𝑥 − 3𝑦 = 4

3𝑥 + (−3𝑥) − 2𝑦 = −3𝑥 − 2

−2𝑦

−2=

−3𝑥 − 2

−2 ; 𝑦 =

3

2𝑥 + 1

𝑥 − 3 (3

2𝑥 + 1) = 4

𝑥 −9

2𝑥 − 3 = 4

−7

2𝑥 − 3 + 3 = 4 + 3; −

7

2𝑥 = 7

(−2

7) (−

7

2𝑥) = (7) (

−2

7) ; 𝑥 = −2

(−2) − 3𝑦 = 4; −2 + 2 − 3𝑦 = 4 + 2 −3𝑦

−3=

6

−3 ; 𝑦 = −2

(−2, −2)

8. 2𝑥 + 5𝑦 = −10 4𝑥 − 5𝑦 = −2

2𝑥 + (−2𝑥) + 5𝑦 = −2𝑥 − 10 5𝑦

5=

−2𝑥 − 10

5; 𝑦 = −

2

5𝑥 − 2

4𝑥 − 5 (−2

5𝑥 − 2) = −2

4𝑥 + 2𝑥 + 10 = −2 ; 6𝑥 + 10 − 10 = −2 − 10

6𝑥 = −12 ; 6𝑥

6=

−12

6 ; 𝑥 = −2

2(−2) + 5𝑦 = −10 ; −4 + 4 + 5𝑦 = −10 + 4 5𝑦

5= −

6

5 ; 𝑦 = −

6

5

(−2, −6

5) 𝑂𝑅 (−2, −1

1

5)

1.6𝑥

1.6=

2

1.6 ; 𝑥 = 1.25

𝑦 = 0.4(1.25) − 1 ; 𝑦 = 0.5 − 1 ; 𝑦 = −0.5

(1.25, −0.5) 𝑂𝑅 (11

4, −

1

2)

25 | P a g e

Answers: Page 11. Remember to check your answers before using the answer key.

1. 2.

3. 4.

3x + 2y = 3

3x +(–4y) = 9

2y=6; −2𝑦

−2=

6

−2 ; y = 3

3x + 2(3) = 3; 3x + (6) + 6 = 3 + 6

3x = 3; 3𝑦

3=

3

3 ; x = 1;

(1, 3)

4x + (3y) = 7

2x + 3y = 5

6x = 12; 6𝑥

6=

−12

6 ; x = 2

2(2)+3y = 5; 4 + 3y = 5 ; 4 + 4 + 3y = 5 + 4

3y = 1 ; 3𝑦

3=

−1

3; 𝑦 = −

1

3

(−2, −1

3)

3x + 5y = 6

(1)[3x + (y)] = (6)(1)

3x + 5y = 6

3x + y = 6

6y = 12; 6𝑦

6=

12

6 ; 𝑦 = 2

3x+5(2)=6 ; 3x+10 + (10) = 6 + (10)

3x = 4 ; 3𝑥

3=

−4

3 ; 𝑥 = −1

1

3

(11

3 , 2)

(2)(5x + 2y) = (7)(2)

x + 4y = 4

10x + (4y) = 14

x + 4y = 4

9x = 18 ; −9𝑥

−9=

−18

−9 ; 𝑥 = 2

(2) + 4y = 4 ; 2 + (2) + 4y = 4 + (2) ; 4y = 6 4𝑦

4=

−6

4 ; 𝑦 = −1

1

2 ; (2, −1

1

2)

+

CHECK: 3(1) + 2(3) = 3; 3 + (6) = 3

3 = 3

3(1) – 4(3) = 9; 3 + 12 = 9

9=9

+

CHECK: 4(2) 3(−1

3) = 7 ; 8 + 1 = 7

7 = 7

2(2) + 3(−1

3) = 5 ; 4 + (1) = 5 ; 5 = 5

+

CHECK: 3(11

3) + 5(2) = 6; 4 + 10 = 6

6 = 6

3(11

3) – (2) = 6; 4 2 = 6

6 = 6

+

CHECK: 5(2) + 2(−11

2) = 7; 10 + (3) = 7

7 = 7

(2) + 4(−11

2) = 4; 2 + (6) = 4

4 = 4

(next page)

5. (1

9) (9x – 18y) = (27)(

1

9)

7x + 2y = 1

x + (2y) = 3

7x + 2y = 1

8x = 4 ; 8𝑥

8=

−4

8 ; 𝑥 = −

1

2

7(−1

2) + 2𝑦 = −1; −

7

2+ 2𝑦 = −1; −

7

2+

7

2+ 2𝑦 = −1 +

7

2

(1

2) (2𝑦) = (

5

2) (

1

2) ; (−

1

2 , 1

1

4 )

CHECK: 9(−1

2) 18(1

1

4) = 27; −

9

2−

45

2= −27

−54

2= −27 ; −27 = −27

7(−1

2) + 2(1

1

4) = 1 ; −

7

2+

5

2= −1 ; −

2

2= −1

1 = 1

+

26 | P a g e

Answers: Page 11

6.

5x 3y = 3

5y + 7x = 4

(5)[5𝑥 + (−3𝑦)] = (−3)(−5)

(3)[7𝑥 + (−5𝑦)] = (−4)(3)

25x + 15y = 15

21x + (15y) = 12

4x = 3; −4𝑥

−4=

3

−4 ; 𝑥 = −

3

4

5 (−3

4) − 3𝑦 = −3 ; −

15

4+

15

4− 3𝑦 = −3 +

15

4

(−1

3) (−3𝑦) = (

3

4) (−

1

3) ; 𝑦 = −

1

4 ; (−

3

4 , −

1

4)

7. 7x + 10y = 8

(−1

4) (8x + 40y) = (48)(−

1

4)

7x + 10y = 8

2x + (10y) = 12

5x = 20 ; 5𝑥

5= −

20

5 ; 𝑥 = −4

7(−4) + 10𝑦 = −8; −28 + 28 + 10𝑦 = −8 + 28 10𝑦

10=

20

10 ; 𝑦 = 2 ; (−4, 2)

8. 9x + 2y = 11

(−2

3) (6x + 3y) = (42) (−

2

3)

9x + 2y = 11

4x + (2y) = 28

13x = 39 ; 13𝑥

13= −

39

13 ; 𝑥 = −3

9(−3) + 2𝑦 = −11 ; −27 + 27 + 2𝑦 = −11 + 27

2𝑦 = 16 ;2𝑦

2=

16

2 ; 𝑦 = 8 ; (−3, 8)

9. 8x + 15y = 2

(5)[2x + (3y)] = (13)(5)

8x + 15y = 2

10x + (15y) = 65

18x = 63 ; 18𝑥

18=

−63

18 ; 𝑥 = −

7

2= −3

1

2

8 (−7

2) + 15𝑦 = 2 ; −28 + 28 + 15𝑦 = 2 + 28

15𝑦 = 30 ; 15𝑦

15=

30

15; 𝑦 = 2

+

CHECK: 5 (−3

4) − 3 (−

1

4) = −3 ; −

15

4+

3

4= −3

−12

4= −3 ; −3 = −3

5 (−1

4) + 7 (−

3

4) = −4 ;

5

4−

21

4= −4

−16

4= −4 ; −4 = −4

+

CHECK: 7(−4) + 10(2) = −8 ; −28 + 20 = −8

−8 = −8

8(−4) + 40(2) = 48 ; −32 + 80 = 48

48 = 48

+

CHECK: 9(−3) + 2(8) = −11 ; −27 + 16 = −11 −11 = −11

−6(−3) + 3(8) = 42 ; 18 + 24 = 42

42 = 42

+

CHECK: 8 (−7

2) + 15(2) = 2; −28 + 30 = 2

2 = 2

2 (−7

2) − 3(2) = −13; −7 − 6 = −13

−13 = −13

27 | P a g e

Answers: Pages 15 – 16 . Please ask for help if you are having difficulty graphing quickly and

correctly.

Write a system of linear inequalities for each of the following.

1. Quadrant II 2. Quadrant IV 3. Quadrant I 4. Quadrant III

x < 0 x > 0 x > 0 x < 0

y > 0 y < 0 y > 0 y < 0

Graph the system of linear inequalities. Do a full check.

Remember to use the original problem when checking.

Check: 1st borderline: (0, 2), (2, 1)

6(0)+4(2) = 8 ; 8 = 8

6(2) +4(1) = 8 ; 12 + 4 = 8; 8 = 8

< and dashed

2nd borderline: (0,1), (1,3) ; (1)=2(0)+1 ; 1=1

(3)=2(1)+1 ; 3 = 2+1 ; 3=3

≥ and solid

Shaded: (2,0) ; (0) ≥ 2(2)+1 ; 0 ≥ 4+1; 0≥3

6(2)+4(0)<8 ; 12 +4 < 8 ; 8 < 8

y

x

y

x

Check: 1st borderline: (0,2),(2,1)

3(0)2(2)=4 ; 4=4

3(2)2(1)=4 ; 62=4 ; 4=4

> and dashed

2nd borderline: (0,1), (1,0) ; (1) = (0) +1 ; 1=1

(0)= (1) + 1 ; 0=0

< and dashed

Shaded (1,1) ; (1)< (1) + 1 ; 1 < 0

3(1) 2(1) > 4 ; 3 + 2 > 4 ; 5 > 4

5. 6𝑥 + 4𝑦 < −8 ; 6𝑥 + (−6𝑥) + 4𝑦 < −6𝑥 − 8 4𝑦

4<

−6𝑥 − 8

4 ; 𝑦 < −

3

2𝑥 − 2

6. 3𝑥 + (−3𝑥) − 2𝑦 > −3𝑥 + 4 −2𝑦

−2>

−3𝑥 + 4

−2 ; 𝑦 <

3

2𝑥 − 2

28 | P a g e

Answers: Pages 15 & 16

7. x = pounds of Gala apples ; y = pounds of Granny Smith apples (It doesn’t matter which one is the

dependent variable.) x + y ≥ 4

x + 1.5y < 6


We will discuss this in class. Carefully think about the problem.

8. x = # of small seashells ; y = # of supersized seashells (It doesn’t matter which one is the dependent

variable.)


We will discuss this in class. Carefully think about the problem.

x + y ≤ 7 1

2𝑥 + 2𝑦 ≥ 9

Write a system for each graph. Use slope-intercept form.

Please ask for help if you are having difficulty writing the inequalities quickly and correctly.

x + (x) + y ≥ x + 4 ; y ≥ x + 4

𝑥 + (𝑥) +3

2𝑦 < −𝑥 + 6

(2

3) (

3

2𝑦) < (−𝑥 + 6) (

2

3)

𝒚 < −𝟐

𝟑𝒙 + 𝟒

y

x

x + (x) + y ≤ x + 7 ; y ≤ x + 7

1

2𝑥 + (−

1

2𝑥) + 2𝑦 ≥ −

1

2𝑥 + 9

(1

2) (2𝑦) ≥ (−

1

2𝑥 + 9) (

1

2)

𝒚 ≥ −𝟏

𝟒𝒙 + 𝟒

𝟏

𝟐

y

x

9. y ≥ 2x + 2

y < 2x – 1 10. 𝑦 ≥ −

1

2𝑥 − 2

𝑦 <1

3𝑥 − 1

11. 𝑦 ≥ 2𝑥 − 3

𝑦 ≥ −1

4𝑥 + 2

12. x < 2

x > 4

cc math i unit 7 systems of equations and...

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