Sam wants to write an equation to represent a proportional relationship with a constant of proportionality of 6. He writes the equation y = 6 + x.
Is he correct. If not, what is the correct equation? Explain your answer. Consider the graph of the equation in your response.
No, Sam is not correct. The correct equation for a proportional relationship with a constant of proportionality of 6 would be y = 6x.
In Sam's equation y = 6 + x, the "+x" term implies that there is an additional constant term being added to the value of y, which would result in a linear relationship rather than a proportional relationship.
If we graph Sam's equation y = 6 + x, it would result in a straight line with a slope of 1 and a y-intercept of 6. This indicates that every time x increases by 1, y will also increase by 1. However, this does not represent a proportional relationship as the constant of proportionality is not 6.
On the other hand, if we graph the correct equation y = 6x, it would result in a straight line passing through the origin (0,0) with a slope of 6. This indicates that for every increase in x by 1, y will increase by 6, maintaining the constant of proportionality of 6. Therefore, y = 6x represents a proportional relationship.
The equation y =
1
5
x represents a proportional relationship. Explain how you can tell the relationship is proportional from the graph of the equation, and how you can find the constant of proportionality.
From the equation, y = (1/5)x, we can tell that the relationship is proportional. This is because the equation is in the form y = kx, where k represents the constant of proportionality.
From the graph of the equation, we can observe that it is a straight line passing through the origin (0,0). This indicates that as x increases, y also increases, maintaining a constant ratio between them.
To find the constant of proportionality, we can examine the slope of the line. In this case, the slope of the line is (1/5). The slope represents the change in y divided by the change in x.
Since the equation is in the form y = kx, we can equate the slope to the constant of proportionality: (1/5) = k. Therefore, the constant of proportionality is 1/5. This means that for every increase in x by 1, y will increase by 1/5, maintaining the proportional relationship.
Jason has applied to be a camp counselor for the summer. The job pays $9 per hour. The equation to represent Jason’s job is y = 9x, where x is the number of hours he works and y is the total amount he earns.
Mia has applied to be a lifeguard for the summer. The lifeguard job is three days a week with hours and pay as shown in the table below.
Tuesday
Thursday
Saturday
Hours worked
6
8
5
Amount paid
$52.50
$70.00
$43.75
Which statement best describes the hourly rates?
The two jobs pay the same hourly rate.
The comparison cannot be made with the information given.
Jason’s camp counselor job pays a higher hourly rate than Mia’s lifeguard job.
Jason’s camp counselor job pays a lower hourly rate than Mia’s lifeguard job.
We can determine the hourly rates for both jobs by dividing the amount paid by the number of hours worked.
For Jason's camp counselor job, the equation y = 9x represents the relationship between the amount paid (y) and the number of hours worked (x). Dividing both sides of the equation by x, we get y/x = 9. Therefore, the hourly rate for Jason's job is $9.
For Mia's lifeguard job, the table provides the amount paid and the number of hours worked for each day. Dividing the amount paid by the number of hours worked for each day, we get the following hourly rates:
Tuesday: $52.50 / 6 = $8.75
Thursday: $70.00 / 8 = $8.75
Saturday: $43.75 / 5 = $8.75
The hourly rate for Mia's lifeguard job is $8.75.
Therefore, Jason's camp counselor job pays a higher hourly rate than Mia's lifeguard job. The correct statement is:
Jason’s camp counselor job pays a higher hourly rate than Mia’s lifeguard job.
Which equation represents a proportional relationship that has a constant of proportionality equal to One-fifth?
y = x minus one-fifth
y = x + one-fifth
y = one-fifth x
y = 5 x
The equation that represents a proportional relationship with a constant of proportionality equal to one-fifth is:
y = (1/5)x
TIME REMAINING
53:53
Rita is developing an equation that will represent the same proportional relationship as the graph.
On a coordinate plane, a line goes through points (0, 0) and (4, 7).
How will the ordered pair help her when she finds the equation?
The product of the coordinates will be a constant term in the equation.
The quotient of the coordinates will be a constant term in the equation.
The product of the coordinates will be a coefficient in the equation.
The quotient of the coordinates will be a coefficient in the equation.
In this case, the quotient of the coordinates (4, 7) would be a constant term in the equation.
The equation to represent the proportional relationship between the points (0, 0) and (4, 7) on a coordinate plane can be written as y = (7/4)x.
The coefficient is the ratio of the y-coordinate (7) to the x-coordinate (4), which is (7/4). This represents the constant rate of change between the two points and is the slope of the line. Therefore, the quotient of the coordinates serves as the coefficient in the equation.
Roni wants to write an equation to represent a proportional relationship that has a constant of proportionality equal to StartFraction 7 over 25 EndFraction. She writes the equation y = x + StartFraction 7 over 25 EndFraction. What error is Roni making?
She should have written y = negative x + StartFraction 7 over 25 EndFraction so that x and y have a constant sum.
She should have written x y = StartFraction 7 over 25 EndFraction so that x and y have a constant product.
She should have written y = StartFraction 7 over 25 EndFraction x so that x and y have a constant quotient.
She should have written y = StartFraction 7 over 25 EndFraction so that y has a constant value.
The error Roni is making is that she should have written the equation as y = (7/25)x.
The equation y = x + (7/25) does not accurately represent a proportional relationship with a constant of proportionality equal to 7/25.
In a proportional relationship, the equation should be in the form of y = kx, where k represents the constant of proportionality. In this specific case, the constant of proportionality is 7/25. Therefore, the correct equation should be y = (7/25)x, where y is directly proportional to x with a constant rate of change of 7/25.
The Hernandez family orders 3 large pizzas. They cut the pizzas so that each pizza has the same number of slices, giving them a total of 24 slices.
The Wilson family also orders several large pizzas from the same pizza restaurant. They also cut the pizzas so that all their pizzas have the same number of slices. For the Wilson family, the equation y = 10x represents the relationship, where x represents the number of pizzas and y represents the number of total slices.
Which statements best describe the pizzas bought by the Hernandez and Wilson families? Select two options.
For the Hernandez family, the equation y = 3x represents the relationship between the number of slices and number of pizzas, where x represents the number of pizzas and y represents the number of total slices.
For each family, a graph of the relationship between number of pizzas, x, and number of slices, y, goes through the point (0, 0) and is a straight line.
The Hernandez family’s slices are smaller than the Wilson family’s slices.
If the Wilson family ordered 3 large pizzas from the same restaurant as the Hernandez family, they would have more slices than the Hernandez family after each family cut their pizzas into slices.
The constant of proportionality is 10 for the Wilson family, but cannot be determined for the Hernandez family.
The correct statements are:
1. For each family, a graph of the relationship between number of pizzas, x, and number of slices, y, goes through the point (0, 0) and is a straight line.
This is true for both families as cutting 0 pizzas would result in 0 slices.
2. If the Wilson family ordered 3 large pizzas from the same restaurant as the Hernandez family, they would have more slices than the Hernandez family after each family cut their pizzas into slices.
This is true because for the Wilson family's equation y = 10x, with x = 3 pizzas, they would have 10 x 3 = 30 slices, which is more than the Hernandez family's 24 slices.
The other statements are incorrect:
1. For the Hernandez family, the equation y = 3x does not represent the relationship between the number of slices and number of pizzas. The correct equation is y = 8x, since there are 8 slices per pizza and they ordered 3 pizzas in total.
2. The size of the slices is not mentioned and cannot be inferred from the given information. No statement can be made regarding the size of the slices for the two families.
3. The constant of proportionality is 8 for the Hernandez family, as each pizza has 8 slices.
What is the constant of proportionality in the equation?
y = 5 x
0
1
4
5
The constant of proportionality in the equation y = 5x is 5.
The fine schedule for overdue books at the county library is modeled by the values in the table.
Library Fines for Overdue Books
Days overdue
Amount of fine
0
0 ¢
2
5 ¢
4
10 ¢
6
15 ¢
8
20 ¢
10
25 ¢
Which equation best models the fine schedule for overdue books?
y = five-halves x, where x is the cost in cents for a book that is y days overdue
y = five-halves x, where y is the cost in cents for a book that is x days overdue
y = 5 x, where x is the cost in cents for a book that is y days overdue
y = 5 x, where y is the cost in cents for a book that is x days overdue
The equation that best models the fine schedule for overdue books is:
y = 5x, where y is the cost in cents for a book that is x days overdue.
In the given table, the cost of the fine (y) is directly proportional to the number of days overdue (x). As each day passes, the fine increases by 5 cents. Therefore, the equation that represents this relationship is y = 5x, where y is the cost in cents and x is the number of days overdue.
Which equation represents a proportional relationship that has a constant of proportionality equal to 2?
y = x + 2
y = StartFraction x over 2 EndFraction
y = 2 x
y = 2
The equation that represents a proportional relationship with a constant of proportionality equal to 2 is:
y = 2x.
In this equation, the value of y is directly proportional to the value of x, with a constant rate of change of 2. This means that every time x increases by 1, y increases by 2, maintaining the constant ratio of 2:1. Therefore, y = 2x represents a proportional relationship with a constant of proportionality equal to 2. The other given equations do not represent this relationship.
Samir works 15 hours for every 19 hours that Mitul works. If x represents the number of hours that Mitul works and y represents the hours that Samir works, which equation correctly models this relationship?
y = StartFraction 15 over 19 EndFraction x
y = StartFraction 19 over 15 EndFraction x
y = 15 x + 19
y = 19 x + 15
The equation that correctly models this relationship is:
y = (15/19)x.
In this equation, y represents the hours that Samir works and x represents the number of hours that Mitul works. This equation shows a proportional relationship between the two variables, where the constant of proportionality is 15/19. This means that for every hour Mitul works (x), Samir will work (y) hours, with the ratio of y to x being 15/19. Therefore, y = (15/19)x is the correct equation that models this relationship.
TIME REMAINING
41:07
Joylin is writing an equation to model the proportional relationship between y, the total cost in dollars of downloading videos from a website, and x, the number of videos downloaded. She knows that the total cost to download 3 videos was $12. Her work to find the equation is shown below.
Joylin’s Work
Step 1
k = StartFraction 3 over 12 EndFraction = 0.25
Step 2
y = 0.25 x
Where did Joylin make her first error?
Joylin divided 3 and 12 instead of multiplying the two numbers.
Joylin computed the decimal equivalent of StartFraction 3 over 12 EndFraction incorrectly.
Joylin used 0.25 as the coefficient of x instead of a product of x and y.
Joylin switched the numerator and the denominator when she found k.
Joylin made her first error in Step 1.
Instead of dividing 3 and 12, she should have multiplied the two numbers to find the constant of proportionality, k.
The correct calculation for Step 1 should be:
k = (3/12) = 0.25
So, Joylin's equation in Step 2 should be:
y = 0.25x
So the correct error is:
Joylin divided 3 and 12 instead of multiplying the two numbers.