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.