Unit 2 Plan - Algebra 1

Title	Assessment
Lesson 1 Planning a Party	School Supply Reward As a reward for achieving their goals, all students in the ninth grade are awarded a pack of school supplies. Write an expression that could represent an estimated cost for the reward. Use at least one variable. State what each part of the expression represents. Choose a variable in your expression. Describe the values that would be reasonable for the quantity that the letter represents. Show Solution Sample responses: $1.50n + 0.08(1.50n)$ . There are $n$ students in the ninth grade. A notebook with the school logo is about \$1.50, and tax is about 8%. $\frac {125}{6} \boldcdot 5 + 6(3) + p$ . There are 125 students in the ninth grade. Suppose a pack of notebooks costs \$5 and has 6 notebooks. A pack of 24 erasers costs about \$3 and 6 packs are needed. A pack of 150 bookmarks costs $p$ dollars. Sample responses: A whole number between 90 and 150 is a reasonable value for the number of students in ninth grade. A number less than $4 but more than $1 is reasonable for the cost of a pack of erasers.
Lesson 2 Writing Equations to Model Relationships (Part 1)	Shirt Colors A school choir needs to make T-shirts for its 75 members and has set aside some money in their budget to pay for them. The members of the choir decided to order from a printing company that charges $3 per shirt, plus a $50 set-up fee for each color to be printed on the shirts. Write an equation that represents the relationship between the number of T-shirts ordered, the number of colors on the shirts, and the total cost of the order. If you use a variable, specify what it represents. In this situation, which quantities do you think can vary? Which might be fixed? Show Solution $75(3) + 50x = D$ (or equivalent). $x$ represents the number of colors on the shirts. $D$ represents the total cost in dollars. Sample response: The cost per shirt and the set-up fee-per-color are fixed (they are set by the printing company). The number of colors on the shirts can vary, so can the total cost, depending on the number of colors being printed.
Lesson 3 Writing Equations to Model Relationships (Part 2)	Labeling Books Clare volunteers at a local library during the summer. Her work includes putting labels on 750 books. How many minutes will she need to finish labeling all books if she takes no breaks and labels: 10 books a minute 15 books a minute Suppose Clare labels the books at a constant speed of $s$ books per minute. Write an equation that represents the relationship between her labeling speed and the number of minutes it would take her to finish labeling. Show Solution 75 minutes 50 minutes $s \boldcdot m = 750$ , or $m = \frac{750}{s}$ , where $m$ is the number of minutes and $s$ is the speed in books per minute.
Lesson 4 Equations and Their Solutions	Box of T-shirts An empty shipping box weighs 250 grams. The box is then filled with T-shirts. Each T-shirt weighs 132.5 grams. The equation $W = 250 + 132.5T$ represents the relationship between the quantities in this situation, where $W$ is the weight, in grams, of the filled box, and $T$ is the number of shirts in the box. Name two possible solutions to the equation $W = 250 + 132.5T$ . What do the solutions mean in this situation? Consider the equation $2,900 = 250 + 132.50T$ . In this situation, what does the solution to this equation tell us? Show Solution Sample response: $T=2$ and $W = 515$ $T= 10$ and $W=1,575$ Each solution tells us the number of T-shirts in the box and the corresponding total weight in grams. It tells us the number of T-shirts in the box that results in a total weight of 2,900 grams.
Lesson 5 Equations and Their Graphs	A Spoonful of Sugar A ceramic sugar bowl weighs 340 grams when empty. It is then filled with sugar. One tablespoon of sugar weighs 12.5 grams. Write an equation to represent the relationship between the total weight of the bowl in grams, $W$ , and the tablespoons of sugar, $T$ . When the sugar bowl is full, it weighs 740 grams. How many tablespoons of sugar can the bowl hold? Plot a point on the graph that represents this situation. Show Solution $W=340 + 12.5T$ , or $340 + 12.5T = 740$ 32 tablespoons. A point plotted near $(32,740)$
Section A Check Section A Checkpoint	Problem 1 A hair stylist job pays $20 per hour and $5 per haircut. Describe any variables needed to write an equation to represent how much the hair stylist makes in a day of work. Choose a letter to represent each variable. For example, “ $E$ is the amount of money earned that day.” Write an equation using the variables that you selected to represent how much the hair stylist makes in a day of work. Show Solution Sample response: $h$ is the number of hours worked in the day, $c$ is the number of haircuts that day, and $E$ is the amount of money earned that day. $20h+5c=E$ Problem 2 The equation $4s+t=100$ represents the number of students, $s$ , and teachers, $t$ , who go on a field trip. What does it mean that $s=22$ and $t=12$ is a solution to the equation? How could you use a graph representing the equation to find another solution? Show Solution Sample response: Substituting those values into the equation produces a true equation. A point on the line representing the graph is a solution.

Lesson 6 Equivalent Equations	Box of Beans and Rice A cardboard box, which weighs 0.6 pound when empty, is filled with 15 bags of beans and a 4-pound bag of rice. The total weight of the box and the contents inside it is 25.6 pounds. One way to represent this situation is with the equation $0.6 + 15b + 4 = 25.6$ . In this situation, what does the solution to the equation represent? Select all equations that are also equivalent to $0.6 + 15b + 4=25.6$ . Equation A: $15b + 4 = 25.6$ Equation B: $15b + 4 = 25$ Equation C: $3(0.6 + 15b + 4) = 76.8$ Equation D: $15b = 25.6$ Equation E: $15b = 21$ Show Solution The weight, in pounds, of one bag of beans B, C, and E
Lesson 7 Explaining Steps for Rewriting Equations	If This, Then That The equation $4(x-2) = 100$ is a true equation for a particular value of $x$ . Explain why $2(x-2) = 50$ is also true for the same value of $x$ . To solve the equation $7.5d=2.5d$ , Lin divides each side by $2.5d$ , and Elena subtracts $2.5d$ from each side. Will both moves lead to the solution? Explain your reasoning. What is the solution? Show Solution Sample response: If $4(x-2)$ and 100 are equal, then multiplying $4(x-2)$ by $\frac12$ and multiplying 100 by $\frac12$ will give two results that are also equal, so $2(x-2)=50$ must be true. No. Sample reasoning: Lin's move won't work. Dividing by $2.5d$ gives $3=1$ , which is not true. Dividing by the variable that we are trying to solve for makes us think that there is no solution. $d=0$
Lesson 10 Connecting Equations to Graphs (Part 1)	Kiran at the Carnival Kiran is spending $12 on games and rides at another carnival, where a game costs $0.25 and a ride costs $1. Write an equation to represent the relationship between the dollar amount Kiran is spending and the number of games, $x$ , and the number of rides, $y$ , that he could purchase tickets for. Which graph represents the relationship between the quantities in this situation? Explain how you know. A Graph of a line, origin O. Horizontal axis, number of games, scale is 0 to 18, by 2’s. Vertical axis, number of rides, scale is 0 to 18, by 2’s. Line starts at 0 comma 12, and passes through 2 comma 4. B Graph of a line, origin O. Horizontal axis, number of games, scale is 0 to 18, by 2’s. Vertical axis, number of rides, scale is 0 to 18, by 2’s. Line starts at 0 comma 12, and passes through 2 comma 4. C Graph of a line, origin O. Horizontal axis, number of games, scale is 0 to 18, by 2’s. Vertical axis, number of rides, scale is 0 to 18, by 2’s. Line starts at 0 comma 12, passes through 8 comma 10, and 16 comma 8. Show Solution $0.25x + y = 12$ Graph C (for $y=12-\frac14 x$ ). Sample explanations: If Kiran plays 0 games, he could get on 12 rides. If he plays 8 games, he could get on 10 rides. Both the points $(0,12)$ and $(8,10)$ are on the line in Graph C. Rearranging the equation into slope-intercept form gives $y=12-\frac14 x$ , so the graph has a slope of $\text-\frac 14$ and intersects the $y$ -axis at 12, which matches Graph C.
Lesson 11 Connecting Equations to Graphs (Part 2)	Features of a Graph Consider the equation $1.5x + 4.5y = 18$ . For each question, explain or show your reasoning. If we graph the equation, what is the slope of the graph? Where does the graph intersect the $y$ -axis? Where does it intersect the $x$ -axis? Show Solution Sample responses: The slope is $\text-\frac13$ . Rearranging the equation to isolate $y$ gives $y=\text- \frac13 x+4$ . The graph intersects the $y$ -axis at $(0,4)$ . The 4 in $y=\text- \frac13 x+4$ tells us where the graph crosses the $y$ -axis. The graph intersects the $x$ -axis at $(12,0)$ . Substituting 0 for $y$ in the original equation and solving for $x$ gives $x=12$ .
Section B Check Section B Checkpoint	Problem 1 Select all of the equations that are equivalent to $5x+4y=20$ . A. $5x=20-4y$ B. $x=\frac{4}{5}y+4$ C. $y=\frac{\text-5}{4}x + 5$ D. $5x + 4y - 6 = 14$ E. $10x + 8y = 20$ Show Solution A, C, D Problem 2 The equation $2x-3y=18$ represents a relationship between $x$ and $y$ . Solve the equation for $y$ (to put it into slope-intercept form). What are these values for the equation? slope: $y$ -intercept: $(\underline{\hspace{.5in}},\underline{\hspace{.5in}})$ Show Solution $y = \frac{2}{3}x - 6, y = \frac{2x-18}{3}$ , or an equivalent equation in which $y$ is isolated. slope: $\frac{2}{3}$ , intercept: $(0,\text{-}6)$

Lesson 12 Writing and Graphing Systems of Linear Equations	Fabric Sale At a fabric store, fabrics are sold by the yard. A dressmaker spent $36.35 on 4.25 yards of silk and cotton fabrics for a dress. Silk is $16.90 per yard and cotton is $4 per yard. Here is a system of equations that represent the constraints in the situation. $\displaystyle \begin{cases} \begin {align} x + \hspace{2.2mm} y &= \hspace{2mm} 4.25\\ 16.90x + 4y &= 36.35 \end{align} \end{cases}$ What does the solution to the system represent? Find the solution to the system of equations. Explain or show your reasoning. Show Solution Sample reasoning: The solution represents the combination of lengths of silk and cotton (in yards) that meet both constraints—they add up to 4.25 yards and the cost of buying both is $36.35. 1.5 yards of silk and 2.75 yards of cotton. Sample reasoning: The graph of the equations intersect at $(1.5, 2.75)$ . Solving by substitution gives $x=1.5$ and $y=2.75$ .
Lesson 13 Solving Systems by Substitution	A System to Solve Solve this system of equations without graphing and show your reasoning: $\begin{cases} \begin {align} 5x + y&=7 \\20x+2&= y \end {align}\end{cases}$ Show Solution $x = \frac15, y = 6$ . Sample strategies: Substituting $20x + 2$ for $y$ in the first equation, $5x+y=7$ , gives $5x + (20x+2) = 7$ , or $25x + 2 = 7$ , which then gives $x=\frac15$ . Substituting $\frac15$ for $x$ in either equation and solving for $y$ gives $y=6$ . The first equation, $5x+y=7$ , can be rearranged to $y = 7 - 5x$ . Substituting $7-5x$ for $y$ in the second equation gives $20x + 2 = 7-5x$ . Solving this equation leads to $x=\frac15$ , and substituting $\frac15$ for $x$ in either equation gives $y=6$ .
Lesson 15 Solving Systems by Elimination (Part 2)	Putting New Equations to Work Tyler buys 5 cups of hot cocoa and 4 pretzels for $18.40. Then he buys another 2 cups of hot cocoa and 4 pretzels for $11.20. Here is a system of equations that represent the quantities and constraints in this situation. $\begin {cases} \begin {align} 5c +4p&=18.40 \\2c+4p&= 11.20\end{align} \end{cases}$ What does the solution to the system, $(c,p)$ , represent in this situation? If we add the second equation to the first equation, we have a new equation: $7c + 8p = 29.60$ . Explain why the same $(c,p)$ pair that is a solution to the two original equations is also a solution to this new equation. Show Solution The values of $c$ and $p$ represent the price of a cup of hot cocoa and the price of a pretzel that Tyler paid for. Sample explanations: The new equation shows the total number of cups of hot cocoa and total number of pretzels on one side, and the total amount Tyler spent on the other side. The price of each cup of hot cocoa and the price of each pretzel haven't changed. $2c + 4p$ and 11.20 are equal, so adding $2c+4p$ to one side of an equation and adding 11.20 to the other side means adding equal amounts to the two sides of an equation. This keeps the two sides equal. The same values of $p$ and $c$ that make the original equations true haven't changed.
Lesson 16 Solving Systems by Elimination (Part 3)	Make Your Move Lin and Priya are working to solve this system of equations. $\begin{cases} \begin {align} \frac13x+2y&=4 \\ x+ \hspace{2mm}y&=\text-3 \end {align} \end{cases}$ Lin's first move is to multiply the first equation by 3. Priya's first move is to multiply the second equation by 2. Explain why either move creates a new equation with the same solutions as the original equation. Whose first move would you choose to do to solve the system? Explain your reasoning. Show Solution Sample responses: Multiplying the two sides of an equation by the same factor creates an equivalent equation, which has the same solution as the original equation. Multiplying the two sides of an equation by the same number keeps the two sides equal, so the solution of the first equation still works for the second one. Sample responses: I would choose Priya's move. Multiplying the second equation by 2 gives $2x + 2y= \text-6$ , it can be subtracted from the first to eliminate the $y$ -variable. I would choose Lin's move. Multiplying the first equation by 3 gives $x+6y=12$ . Subtracting the second equation from this equation eliminates the $x$ -variable. Either person's move would work. Priya's move eliminates the $y$ -variable and Lin's move eliminates the $x$ -variable.
Lesson 17 Systems of Linear Equations and Their Solutions	No Graphs, No Problem Mai is given these two systems of linear equations to solve: System 1: $\begin {cases} \begin{align}5x +\hspace{2.2mm}y &=13\\20x + 4y &=64\end{align} \end{cases}$ System 2: $\begin {cases} 5x +y =13\\20x =52 - 4y \end{cases}$ She analyzed them for a moment, and then—without graphing the equations—said, "I got it! One of the systems has no solution, and the other has infinitely many solutions!" Mai is right! Which system has no solution, and which one has many solutions? Explain or show how you know (without graphing the equations). Show Solution The first system has no solutions, and the second system has infinitely many solutions. Sample explanations: Multiplying the first equation by 4 gives $20x + 4y=52$ . Subtracting the second equation from the first equation gives $0 = \text-12$ , which is a false equation and tells us that the system has no solutions. $20x + 4y$ (in the second equation) is 4 times $5x + y$ (in the first equation), but 64 is not 4 times 13. This means there is no pair of $x$ - and $y$ -values that could make both equations simultaneously true. If we isolate $y$ in the first equation, we have $y = 13-5x$ . If we do the same with the second equation, we have $4y = 64-20x$ or $y = 16 - 5x$ . The graphs of the two equations have the same slope (-5), and they intersect the $y$ -axis at different points (13 and 16), so the lines are parallel and have no points (solutions) in common. The second system has infinitely many solutions. Sample explanations: Multiplying the first equation by 4 gives $20x + 4y=52$ . Rearranging the second equation so that the variables are on the left side also gives $20x + 4y=52$ . The two equations are identical, so they have the same solutions. The second equation can be rearranged to $20x + 4y=52$ . We can see that it is 4 times the first equation, which means the two equations, $5x + y = 13$ and $20x + 4y =52$ , are equivalent and have all the same solutions. If we isolate $y$ in each equation, we'd have the exact same equation, $y=13 - 5x$ , so they have all the same solutions.
Section C Check Section C Checkpoint	Problem 1 Solve the system without graphing. $\begin {cases} 3x + 5y = \text{-}9\\ 6x - 7y = 33 \end{cases}$ To solve, you wrote at least one new equation. Explain why this is allowed when solving a system. Explain how you could use a graph to check your answer. Show Solution $x = 2, y = \text{-}3$ Sample response: It is an equivalent equation, so it maintains the solution of the system. Sample response: Graph the two lines and look for the intersection at $(2,\text{-}3)$ . Problem 2 In a system of linear equations, the two equations have the same slope. What information do you know about how many solutions the system has? Explain your reasoning. Show Solution Sample response: There are either 0 or infinitely many solutions. Because the lines have the same slope, they are either parallel (0 solutions) or equivalent equations (infinitely many solutions).

Lesson 1

Planning a Party

School Supply Reward

As a reward for achieving their goals, all students in the ninth grade are awarded a pack of school supplies.

Write an expression that could represent an estimated cost for the reward. Use at least one variable. State what each part of the expression represents.
Choose a variable in your expression. Describe the values that would be reasonable for the quantity that the letter represents.

Show Solution

Sample responses:
- $1.50n + 0.08(1.50n)$ . There are $n$ students in the ninth grade. A notebook with the school logo is about \$1.50, and tax is about 8%.
- $\frac {125}{6} \boldcdot 5 + 6(3) + p$ . There are 125 students in the ninth grade. Suppose a pack of notebooks costs \$5 and has 6 notebooks. A pack of 24 erasers costs about \$3 and 6 packs are needed. A pack of 150 bookmarks costs $p$ dollars.
Sample responses:
- A whole number between 90 and 150 is a reasonable value for the number of students in ninth grade.
- A number less than $4 but more than $1 is reasonable for the cost of a pack of erasers.

Lesson 2

Writing Equations to Model Relationships (Part 1)

Shirt Colors

A school choir needs to make T-shirts for its 75 members and has set aside some money in their budget to pay for them. The members of the choir decided to order from a printing company that charges $3 per shirt, plus a $50 set-up fee for each color to be printed on the shirts.

Write an equation that represents the relationship between the number of T-shirts ordered, the number of colors on the shirts, and the total cost of the order. If you use a variable, specify what it represents.
In this situation, which quantities do you think can vary? Which might be fixed?

Show Solution

$75(3) + 50x = D$ (or equivalent). $x$ represents the number of colors on the shirts. $D$ represents the total cost in dollars.
Sample response: The cost per shirt and the set-up fee-per-color are fixed (they are set by the printing company). The number of colors on the shirts can vary, so can the total cost, depending on the number of colors being printed.

Lesson 3

Writing Equations to Model Relationships (Part 2)

Labeling Books

Clare volunteers at a local library during the summer. Her work includes putting labels on 750 books.

How many minutes will she need to finish labeling all books if she takes no breaks and labels:
1. 10 books a minute
2. 15 books a minute
Suppose Clare labels the books at a constant speed of $s$ books per minute. Write an equation that represents the relationship between her labeling speed and the number of minutes it would take her to finish labeling.

Show Solution

1. 75 minutes
2. 50 minutes
$s \boldcdot m = 750$ , or $m = \frac{750}{s}$ , where $m$ is the number of minutes and $s$ is the speed in books per minute.

Lesson 4

Equations and Their Solutions

Box of T-shirts

An empty shipping box weighs 250 grams. The box is then filled with T-shirts. Each T-shirt weighs 132.5 grams.

The equation $W = 250 + 132.5T$ represents the relationship between the quantities in this situation, where $W$ is the weight, in grams, of the filled box, and $T$ is the number of shirts in the box.

Name two possible solutions to the equation $W = 250 + 132.5T$ . What do the solutions mean in this situation?
Consider the equation $2,900 = 250 + 132.50T$ . In this situation, what does the solution to this equation tell us?

Show Solution

Sample response:
- $T=2$ and $W = 515$
- $T= 10$ and $W=1,575$
- Each solution tells us the number of T-shirts in the box and the corresponding total weight in grams.
It tells us the number of T-shirts in the box that results in a total weight of 2,900 grams.

Lesson 5

Equations and Their Graphs

A Spoonful of Sugar

A ceramic sugar bowl weighs 340 grams when empty. It is then filled with sugar. One tablespoon of sugar weighs 12.5 grams.

Write an equation to represent the relationship between the total weight of the bowl in grams, $W$ , and the tablespoons of sugar, $T$ .
When the sugar bowl is full, it weighs 740 grams. How many tablespoons of sugar can the bowl hold? Plot a point on the graph that represents this situation.

Show Solution

$W=340 + 12.5T$ , or $340 + 12.5T = 740$
32 tablespoons. A point plotted near $(32,740)$

Section A Check

Section A Checkpoint

Problem 1

A hair stylist job pays $20 per hour and $5 per haircut.

Describe any variables needed to write an equation to represent how much the hair stylist makes in a day of work. Choose a letter to represent each variable. For example, “ $E$ is the amount of money earned that day.”
Write an equation using the variables that you selected to represent how much the hair stylist makes in a day of work.

Show Solution

Sample response:

$h$ is the number of hours worked in the day, $c$ is the number of haircuts that day, and $E$ is the amount of money earned that day.
$20h+5c=E$

Problem 2

The equation

4s+t=100

represents the number of students,

s

, and teachers,

t

, who go on a field trip.

What does it mean that $s=22$ and $t=12$ is a solution to the equation?
How could you use a graph representing the equation to find another solution?

Show Solution

Sample response:

Substituting those values into the equation produces a true equation.
A point on the line representing the graph is a solution.

Lesson 6

Equivalent Equations

Box of Beans and Rice

A cardboard box, which weighs 0.6 pound when empty, is filled with 15 bags of beans and a 4-pound bag of rice. The total weight of the box and the contents inside it is 25.6 pounds. One way to represent this situation is with the equation $0.6 + 15b + 4 = 25.6$ .

In this situation, what does the solution to the equation represent?
Select all equations that are also equivalent to $0.6 + 15b + 4=25.6$ .
- Equation A: $15b + 4 = 25.6$
- Equation B: $15b + 4 = 25$
- Equation C: $3(0.6 + 15b + 4) = 76.8$
- Equation D: $15b = 25.6$
- Equation E: $15b = 21$

Show Solution

The weight, in pounds, of one bag of beans
B, C, and E

Lesson 7

Explaining Steps for Rewriting Equations

If This, Then That

The equation $4(x-2) = 100$ is a true equation for a particular value of $x$ . Explain why $2(x-2) = 50$ is also true for the same value of $x$ .
To solve the equation $7.5d=2.5d$ , Lin divides each side by $2.5d$ , and Elena subtracts $2.5d$ from each side.
1. Will both moves lead to the solution? Explain your reasoning.
2. What is the solution?

Show Solution

Sample response: If $4(x-2)$ and 100 are equal, then multiplying $4(x-2)$ by $\frac12$ and multiplying 100 by $\frac12$ will give two results that are also equal, so $2(x-2)=50$ must be true.
1. No. Sample reasoning: Lin's move won't work. Dividing by $2.5d$ gives $3=1$ , which is not true. Dividing by the variable that we are trying to solve for makes us think that there is no solution.
2. $d=0$

Lesson 10

Connecting Equations to Graphs (Part 1)

Kiran at the Carnival

Kiran is spending $12 on games and rides at another carnival, where a game costs $0.25 and a ride costs $1.

Write an equation to represent the relationship between the dollar amount Kiran is spending and the number of games, $x$ , and the number of rides, $y$ , that he could purchase tickets for.
Which graph represents the relationship between the quantities in this situation? Explain how you know.

A

Graph of a line, origin O. Horizontal axis, number of games, scale is 0 to 18, by 2’s. Vertical axis, number of rides, scale is 0 to 18, by 2’s. Line starts at 0 comma 12, and passes through 2 comma 4.

B

Graph of a line, origin O. Horizontal axis, number of games, scale is 0 to 18, by 2’s. Vertical axis, number of rides, scale is 0 to 18, by 2’s. Line starts at 0 comma 12, and passes through 2 comma 4.

C

Graph of a line, origin O. Horizontal axis, number of games, scale is 0 to 18, by 2’s. Vertical axis, number of rides, scale is 0 to 18, by 2’s. Line starts at 0 comma 12, passes through 8 comma 10, and 16 comma 8.

Show Solution

$0.25x + y = 12$
Graph C (for $y=12-\frac14 x$ ). Sample explanations:
- If Kiran plays 0 games, he could get on 12 rides. If he plays 8 games, he could get on 10 rides. Both the points $(0,12)$ and $(8,10)$ are on the line in Graph C.
- Rearranging the equation into slope-intercept form gives $y=12-\frac14 x$ , so the graph has a slope of $\text-\frac 14$ and intersects the $y$ -axis at 12, which matches Graph C.

Lesson 11

Connecting Equations to Graphs (Part 2)

Features of a Graph

Consider the equation $1.5x + 4.5y = 18$ . For each question, explain or show your reasoning.

If we graph the equation, what is the slope of the graph?
Where does the graph intersect the $y$ -axis?
Where does it intersect the $x$ -axis?

Show Solution

Sample responses:

The slope is $\text-\frac13$ . Rearranging the equation to isolate $y$ gives $y=\text- \frac13 x+4$ .
The graph intersects the $y$ -axis at $(0,4)$ . The 4 in $y=\text- \frac13 x+4$ tells us where the graph crosses the $y$ -axis.
The graph intersects the $x$ -axis at $(12,0)$ . Substituting 0 for $y$ in the original equation and solving for $x$ gives $x=12$ .

Section B Check

Section B Checkpoint

Problem 1

Select all of the equations that are equivalent to

5x+4y=20

.

A.

5x=20-4y

B.

x=\frac{4}{5}y+4

C.

y=\frac{\text-5}{4}x + 5

D.

5x + 4y - 6 = 14

E.

10x + 8y = 20

Show Solution

A, C, D

Problem 2

The equation

2x-3y=18

represents a relationship between

x

and

y

.

Solve the equation for $y$ (to put it into slope-intercept form).
What are these values for the equation?
- slope:
- $y$ -intercept: $(\underline{\hspace{.5in}},\underline{\hspace{.5in}})$

Show Solution

$y = \frac{2}{3}x - 6, y = \frac{2x-18}{3}$ , or an equivalent equation in which $y$ is isolated.
slope: $\frac{2}{3}$ , intercept: $(0,\text{-}6)$

Lesson 12

Writing and Graphing Systems of Linear Equations

Fabric Sale

At a fabric store, fabrics are sold by the yard. A dressmaker spent $36.35 on 4.25 yards of silk and cotton fabrics for a dress. Silk is $16.90 per yard and cotton is $4 per yard.

Here is a system of equations that represent the constraints in the situation.

$\displaystyle \begin{cases} \begin {align} x + \hspace{2.2mm} y &= \hspace{2mm} 4.25\\ 16.90x + 4y &= 36.35 \end{align} \end{cases}$

What does the solution to the system represent?
Find the solution to the system of equations. Explain or show your reasoning.

Show Solution

Sample reasoning: The solution represents the combination of lengths of silk and cotton (in yards) that meet both constraints—they add up to 4.25 yards and the cost of buying both is $36.35.
1.5 yards of silk and 2.75 yards of cotton. Sample reasoning:
- The graph of the equations intersect at $(1.5, 2.75)$ .
- Solving by substitution gives $x=1.5$ and $y=2.75$ .

Lesson 13

Solving Systems by Substitution

A System to Solve

Solve this system of equations without graphing and show your reasoning:

$\begin{cases} \begin {align} 5x + y&=7 \\20x+2&= y \end {align}\end{cases}$

Show Solution

$x = \frac15, y = 6$ . Sample strategies:

Substituting $20x + 2$ for $y$ in the first equation, $5x+y=7$ , gives $5x + (20x+2) = 7$ , or $25x + 2 = 7$ , which then gives $x=\frac15$ . Substituting $\frac15$ for $x$ in either equation and solving for $y$ gives $y=6$ .
The first equation, $5x+y=7$ , can be rearranged to $y = 7 - 5x$ . Substituting $7-5x$ for $y$ in the second equation gives $20x + 2 = 7-5x$ . Solving this equation leads to $x=\frac15$ , and substituting $\frac15$ for $x$ in either equation gives $y=6$ .

Lesson 15

Solving Systems by Elimination (Part 2)

Putting New Equations to Work

Tyler buys 5 cups of hot cocoa and 4 pretzels for $18.40. Then he buys another 2 cups of hot cocoa and 4 pretzels for $11.20.

Here is a system of equations that represent the quantities and constraints in this situation.

$\begin {cases} \begin {align} 5c +4p&=18.40 \\2c+4p&= 11.20\end{align} \end{cases}$

What does the solution to the system, $(c,p)$ , represent in this situation?
If we add the second equation to the first equation, we have a new equation: $7c + 8p = 29.60$ .

Explain why the same $(c,p)$ pair that is a solution to the two original equations is also a solution to this new equation.

Show Solution

The values of $c$ and $p$ represent the price of a cup of hot cocoa and the price of a pretzel that Tyler paid for.
Sample explanations:
- The new equation shows the total number of cups of hot cocoa and total number of pretzels on one side, and the total amount Tyler spent on the other side. The price of each cup of hot cocoa and the price of each pretzel haven't changed.
- $2c + 4p$ and 11.20 are equal, so adding $2c+4p$ to one side of an equation and adding 11.20 to the other side means adding equal amounts to the two sides of an equation. This keeps the two sides equal. The same values of $p$ and $c$ that make the original equations true haven't changed.

Lesson 16

Solving Systems by Elimination (Part 3)

Make Your Move

Lin and Priya are working to solve this system of equations.

$\begin{cases} \begin {align} \frac13x+2y&=4 \\ x+ \hspace{2mm}y&=\text-3 \end {align} \end{cases}$

Lin's first move is to multiply the first equation by 3.

Priya's first move is to multiply the second equation by 2.

Explain why either move creates a new equation with the same solutions as the original equation.
Whose first move would you choose to do to solve the system? Explain your reasoning.

Show Solution

Sample responses:
- Multiplying the two sides of an equation by the same factor creates an equivalent equation, which has the same solution as the original equation.
- Multiplying the two sides of an equation by the same number keeps the two sides equal, so the solution of the first equation still works for the second one.
Sample responses:
- I would choose Priya's move. Multiplying the second equation by 2 gives $2x + 2y= \text-6$ , it can be subtracted from the first to eliminate the $y$ -variable.
- I would choose Lin's move. Multiplying the first equation by 3 gives $x+6y=12$ . Subtracting the second equation from this equation eliminates the $x$ -variable.
- Either person's move would work. Priya's move eliminates the $y$ -variable and Lin's move eliminates the $x$ -variable.

Lesson 17

Systems of Linear Equations and Their Solutions

No Graphs, No Problem

Mai is given these two systems of linear equations to solve:

System 1:

$\begin {cases} \begin{align}5x +\hspace{2.2mm}y &=13\\20x + 4y &=64\end{align} \end{cases}$

System 2:

$\begin {cases} 5x +y =13\\20x =52 - 4y \end{cases}$

She analyzed them for a moment, and then—without graphing the equations—said, "I got it! One of the systems has no solution, and the other has infinitely many solutions!" Mai is right!

Which system has no solution, and which one has many solutions? Explain or show how you know (without graphing the equations).

Show Solution

The first system has no solutions, and the second system has infinitely many solutions. Sample explanations:

Multiplying the first equation by 4 gives $20x + 4y=52$ . Subtracting the second equation from the first equation gives $0 = \text-12$ , which is a false equation and tells us that the system has no solutions.
$20x + 4y$ (in the second equation) is 4 times $5x + y$ (in the first equation), but 64 is not 4 times 13. This means there is no pair of $x$ - and $y$ -values that could make both equations simultaneously true.
If we isolate $y$ in the first equation, we have $y = 13-5x$ . If we do the same with the second equation, we have $4y = 64-20x$ or $y = 16 - 5x$ . The graphs of the two equations have the same slope (-5), and they intersect the $y$ -axis at different points (13 and 16), so the lines are parallel and have no points (solutions) in common.

The second system has infinitely many solutions. Sample explanations:

Multiplying the first equation by 4 gives $20x + 4y=52$ . Rearranging the second equation so that the variables are on the left side also gives $20x + 4y=52$ . The two equations are identical, so they have the same solutions.
The second equation can be rearranged to $20x + 4y=52$ . We can see that it is 4 times the first equation, which means the two equations, $5x + y = 13$ and $20x + 4y =52$ , are equivalent and have all the same solutions.
If we isolate $y$ in each equation, we'd have the exact same equation, $y=13 - 5x$ , so they have all the same solutions.

Section C Check

Section C Checkpoint

Problem 1

Solve the system without graphing.

$\begin {cases} 3x + 5y = \text{-}9\\ 6x - 7y = 33 \end{cases}$
To solve, you wrote at least one new equation. Explain why this is allowed when solving a system.
Explain how you could use a graph to check your answer.

Show Solution

$x = 2, y = \text{-}3$
Sample response: It is an equivalent equation, so it maintains the solution of the system.
Sample response: Graph the two lines and look for the intersection at $(2,\text{-}3)$ .

Problem 2

In a system of linear equations, the two equations have the same slope. What information do you know about how many solutions the system has? Explain your reasoning.

Show Solution

Sample response: There are either 0 or infinitely many solutions. Because the lines have the same slope, they are either parallel (0 solutions) or equivalent equations (infinitely many solutions).

Unit 2 Linear Equations And Systems — Unit Plan