There are 48 inches in 4 feet, which is represented by the equation 4y=48x. Determine the number of inches per foot.(1 point)
inches
.
11 answers
12 inches per foot.
Use the image to answer the question.
An illustration shows a coordinate plane with 4 quadrants. The x-axis ranges from negative 9 to 9 in one unit increments, and the y-axis ranges from negative 11 to 11 in one unit increments. A line is graphed on the plane. An upward slanting line passes through points plotted at left parenthesis 1 comma 5 right parenthesis and left parenthesis 2 comma 10 right parenthesis.
The graph shows the proportional relationship. Derive the equation of the line y=mx through the origin.
(1 point)
An illustration shows a coordinate plane with 4 quadrants. The x-axis ranges from negative 9 to 9 in one unit increments, and the y-axis ranges from negative 11 to 11 in one unit increments. A line is graphed on the plane. An upward slanting line passes through points plotted at left parenthesis 1 comma 5 right parenthesis and left parenthesis 2 comma 10 right parenthesis.
The graph shows the proportional relationship. Derive the equation of the line y=mx through the origin.
(1 point)
To find the equation of the line passing through the points (1,5) and (2,10) in the form y = mx, we first need to find the slope (m) of the line.
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Using the points (1,5) and (2,10):
m = (10 - 5) / (2 - 1)
m = 5
Now that we have the slope (m = 5), the equation of the line passing through the origin (0,0) and having this slope is simply y = 5x.
Therefore, the equation of the line through the origin is y = 5x.
The slope of a line passing through two points (x₁, y₁) and (x₂, y₂) is given by the formula:
m = (y₂ - y₁) / (x₂ - x₁)
Using the points (1,5) and (2,10):
m = (10 - 5) / (2 - 1)
m = 5
Now that we have the slope (m = 5), the equation of the line passing through the origin (0,0) and having this slope is simply y = 5x.
Therefore, the equation of the line through the origin is y = 5x.
Graph the proportional relationship y=2x by plotting points.(1 point)
Responses
A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 2 right parenthesis, and left parenthesis 2 comma 4 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 2 right parenthesis, and left parenthesis 2 comma 4 right parenthesis.
A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins two plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis and left parenthesis 1 comma 4 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins two plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis and left parenthesis 1 comma 4 right parenthesis.
A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 3 right parenthesis, and left parenthesis 2 comma 6 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 3 right parenthesis, and left parenthesis 2 comma 6 right parenthesis.
Responses
A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 2 right parenthesis, and left parenthesis 2 comma 4 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 2 right parenthesis, and left parenthesis 2 comma 4 right parenthesis.
A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins two plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis and left parenthesis 1 comma 4 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins two plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis and left parenthesis 1 comma 4 right parenthesis.
A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 3 right parenthesis, and left parenthesis 2 comma 6 right parenthesis.
Image with alt text: A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins three plotted points and extends beyond. The coordinates of the plotted points are: left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 3 right parenthesis, and left parenthesis 2 comma 6 right parenthesis.
The correct graph for the proportional relationship y = 2x is the second one described in your responses: "A coordinate plane ranges from negative 9 to 9 in unit increments on the x-axis and from 11 to negative 11 in unit increments on the y-axis. A line with arrows at both the ends joins two plotted points and extends beyond. The coordinates of the plotted points are: (0, 0) and (1, 4)."
This graph correctly represents the points (0, 0) and (1, 4) on the line y = 2x as they lie on the line and show the relationship between x and y where y is twice the value of x.
This graph correctly represents the points (0, 0) and (1, 4) on the line y = 2x as they lie on the line and show the relationship between x and y where y is twice the value of x.
Use the image to answer the question.
An illustration shows Quadrant 1 of a coordinate plane. The x-axis is labeled Ounces and ranges from 0 to 15 in 3 unit increments. The y-axis ranges from 0 to 153 in 9 unit increments. The graph is titled Calories per Ounce. A line connects five points plotted on the graph. The coordinates of the plotted points are left parenthesis 0 comma 0 right parenthesis, left parenthesis 3 comma 36 right parenthesis, left parenthesis 6 comma 72 right parenthesis, left parenthesis 9 comma 108 right parenthesis, and left parenthesis 12 comma 144 right parenthesis.
The proportional relationship between calories and ounces of soda is shown in the graph. How many calories are in 1 ounce?
(1 point)
There are
calories in 1 ounce of soda.
An illustration shows Quadrant 1 of a coordinate plane. The x-axis is labeled Ounces and ranges from 0 to 15 in 3 unit increments. The y-axis ranges from 0 to 153 in 9 unit increments. The graph is titled Calories per Ounce. A line connects five points plotted on the graph. The coordinates of the plotted points are left parenthesis 0 comma 0 right parenthesis, left parenthesis 3 comma 36 right parenthesis, left parenthesis 6 comma 72 right parenthesis, left parenthesis 9 comma 108 right parenthesis, and left parenthesis 12 comma 144 right parenthesis.
The proportional relationship between calories and ounces of soda is shown in the graph. How many calories are in 1 ounce?
(1 point)
There are
calories in 1 ounce of soda.
Based on the given graph and the plotted points (0,0), (3,36), (6,72), (9,108), and (12,144), we can observe that the relationship between the number of ounces and the corresponding calories is in a linear proportion.
The consistent increase in calories with respect to ounces indicates a linear relationship where the slope (m) of this line is constant. To find out how many calories are in 1 ounce, we can look at the slope of the line connecting the plotted points.
Calculating the slope:
m = (Change in y) / (Change in x) = (144 - 0) / (12 - 0) = 144 / 12 = 12
This means that for each additional ounce of soda, there are 12 calories. Therefore, there are 12 calories in 1 ounce of soda.
The consistent increase in calories with respect to ounces indicates a linear relationship where the slope (m) of this line is constant. To find out how many calories are in 1 ounce, we can look at the slope of the line connecting the plotted points.
Calculating the slope:
m = (Change in y) / (Change in x) = (144 - 0) / (12 - 0) = 144 / 12 = 12
This means that for each additional ounce of soda, there are 12 calories. Therefore, there are 12 calories in 1 ounce of soda.
Use the image to answer the question.
An illustration shows two graphs depicting earnings per hour for employees 1 and 2. The first graph shows earnings per hour for employee 1. The horizontal axis shows hours ranging from 0 to 8 in increments of 2. The vertical axis shows earnings in dollars ranging from 0 to 80 in increments of 5. Four data points are plotted on the graph at left parenthesis 0 comma 0 right parenthesis, left parenthesis 2 comma 25 right parenthesis, left parenthesis 4 comma 50 right parenthesis, and left parenthesis 6 comma 75 right parenthesis. A line connects all the data points. The second graph shows earnings per hour for employee 2. The horizontal axis shows hours ranging from 0 to 6 in increments of 1. The vertical axis shows earnings in dollars ranging from 0 to 80 in increments of 5. Four data points are plotted on the graph at left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 15 right parenthesis, left parenthesis 3 comma 45 right parenthesis, and left parenthesis 5 comma 75 right parenthesis. A line connects all the data points.
The earnings for Employee 1 and Employee 2 are displayed in the following graphs. Which employee earns more per hour?
(1 point)
Employee
earns more per hour.
An illustration shows two graphs depicting earnings per hour for employees 1 and 2. The first graph shows earnings per hour for employee 1. The horizontal axis shows hours ranging from 0 to 8 in increments of 2. The vertical axis shows earnings in dollars ranging from 0 to 80 in increments of 5. Four data points are plotted on the graph at left parenthesis 0 comma 0 right parenthesis, left parenthesis 2 comma 25 right parenthesis, left parenthesis 4 comma 50 right parenthesis, and left parenthesis 6 comma 75 right parenthesis. A line connects all the data points. The second graph shows earnings per hour for employee 2. The horizontal axis shows hours ranging from 0 to 6 in increments of 1. The vertical axis shows earnings in dollars ranging from 0 to 80 in increments of 5. Four data points are plotted on the graph at left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 15 right parenthesis, left parenthesis 3 comma 45 right parenthesis, and left parenthesis 5 comma 75 right parenthesis. A line connects all the data points.
The earnings for Employee 1 and Employee 2 are displayed in the following graphs. Which employee earns more per hour?
(1 point)
Employee
earns more per hour.
To determine which employee earns more per hour, we need to compare the slopes of the lines representing the earnings per hour for Employee 1 and Employee 2 in the graphs.
Starting with Employee 1:
The data points provided for Employee 1 are: (0,0), (2,25), (4,50), and (6,75).
For Employee 1:
- Slope = Change in Earnings / Change in Hours
- Slope = (75 - 0) / (6 - 0) = 75 / 6 = 12.5
Therefore, the slope for Employee 1 is 12.5.
Moving on to Employee 2:
The data points provided for Employee 2 are: (0,0), (1,15), (3,45), and (5,75).
For Employee 2:
- Slope = Change in Earnings / Change in Hours
- Slope = (75 - 0) / (5 - 0) = 75 / 5 = 15
Therefore, the slope for Employee 2 is 15.
Comparing the slopes, Employee 2 earns more per hour as the slope for Employee 2 is higher than the slope for Employee 1. Thus, Employee 2 earns more per hour.
Starting with Employee 1:
The data points provided for Employee 1 are: (0,0), (2,25), (4,50), and (6,75).
For Employee 1:
- Slope = Change in Earnings / Change in Hours
- Slope = (75 - 0) / (6 - 0) = 75 / 6 = 12.5
Therefore, the slope for Employee 1 is 12.5.
Moving on to Employee 2:
The data points provided for Employee 2 are: (0,0), (1,15), (3,45), and (5,75).
For Employee 2:
- Slope = Change in Earnings / Change in Hours
- Slope = (75 - 0) / (5 - 0) = 75 / 5 = 15
Therefore, the slope for Employee 2 is 15.
Comparing the slopes, Employee 2 earns more per hour as the slope for Employee 2 is higher than the slope for Employee 1. Thus, Employee 2 earns more per hour.
Use the image to answer the question.
An illustration shows a coordinate plane with the x-axis ranging from negative 9 to 9 in one unit increments, and the y-axis ranging from negative 11 to 11 in one unit increments. A solid line passes through four plotted points and extends beyond. A solid line with arrows at both ends passes through points with coordinates left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 3 right parenthesis, left parenthesis 2 comma 6 right parenthesis, and left parenthesis 3 comma 9 right parenthesis. The solid line forms the hypotenuse for two triangles. The first triangle is marked with a dotted line and has vertices with coordinates left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 0 right parenthesis, and left parenthesis 1 comma 3 right parenthesis. The base is labeled as 1, and the perpendicular height on the right is labeled as 3. The second triangle is marked with a dashed line and has vertices with coordinates left parenthesis 1 comma 3 right parenthesis, left parenthesis 3 comma 3 right parenthesis, and left parenthesis 3 comma 9 right parenthesis. The base is labeled as 2, and the perpendicular height on the right is labeled as 6.
Use similar triangles to determine the slope of the line.
(1 point)
The slope is
An illustration shows a coordinate plane with the x-axis ranging from negative 9 to 9 in one unit increments, and the y-axis ranging from negative 11 to 11 in one unit increments. A solid line passes through four plotted points and extends beyond. A solid line with arrows at both ends passes through points with coordinates left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 3 right parenthesis, left parenthesis 2 comma 6 right parenthesis, and left parenthesis 3 comma 9 right parenthesis. The solid line forms the hypotenuse for two triangles. The first triangle is marked with a dotted line and has vertices with coordinates left parenthesis 0 comma 0 right parenthesis, left parenthesis 1 comma 0 right parenthesis, and left parenthesis 1 comma 3 right parenthesis. The base is labeled as 1, and the perpendicular height on the right is labeled as 3. The second triangle is marked with a dashed line and has vertices with coordinates left parenthesis 1 comma 3 right parenthesis, left parenthesis 3 comma 3 right parenthesis, and left parenthesis 3 comma 9 right parenthesis. The base is labeled as 2, and the perpendicular height on the right is labeled as 6.
Use similar triangles to determine the slope of the line.
(1 point)
The slope is
To determine the slope of the line using similar triangles, we can compare the ratios of the corresponding sides of the two triangles.
From the information given in the question, the right triangle with vertices (0,0), (1,0), and (1,3) has base 1 and perpendicular height 3.
The larger triangle with vertices (1,3), (3,3), and (3,9) has base 2 (as it is two units wide) and perpendicular height 6.
Since the triangles are similar, the ratio of the base to the height will be the same for both triangles. Therefore, the slope of the line will be given by the ratio of the change in y to the change in x for either triangle.
For the first triangle:
Slope = Change in y / Change in x = 3 / 1 = 3
For the second triangle:
Slope = Change in y / Change in x = 6 / 2 = 3
So, the slope of the line passing through the plotted points is 3.
From the information given in the question, the right triangle with vertices (0,0), (1,0), and (1,3) has base 1 and perpendicular height 3.
The larger triangle with vertices (1,3), (3,3), and (3,9) has base 2 (as it is two units wide) and perpendicular height 6.
Since the triangles are similar, the ratio of the base to the height will be the same for both triangles. Therefore, the slope of the line will be given by the ratio of the change in y to the change in x for either triangle.
For the first triangle:
Slope = Change in y / Change in x = 3 / 1 = 3
For the second triangle:
Slope = Change in y / Change in x = 6 / 2 = 3
So, the slope of the line passing through the plotted points is 3.