Question Two partners split profits evenly. Find the equation, graph, and determine if proportional for $x$ (profit) and $y$ (each partner's share).

Studdy Solution

STEP 1

Assumptions
1. The table provides pairs of $x$ (profit) and $y$ (amount received by each partner).
2. The profits are split evenly between the two business partners.
3. We are to assume that the relationship between $x$ and $y$ can be represented by a function.

STEP 2

To write an equation for the function shown in the table, we need to determine the relationship between $x$ and $y$ . We can observe that for every increase in $x$ , there is a corresponding increase in $y$ .

STEP 3

By examining the table, we can see that when $x$ increases by 2, $y$ increases by 1. This suggests a linear relationship between $x$ and $y$ .

STEP 4

To find the equation of the line, we can use the slope-intercept form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

STEP 5

Since the y-intercept is the value of $y$ when $x = 0$ , we can see from the table that $b = 0$ .

STEP 6

To find the slope $m$ , we can use the change in $y$ over the change in $x$ between any two points from the table.

STEP 7

Using the points (2, 1) and (4, 2), we calculate the slope as follows:
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 1}{4 - 2} = \frac{1}{2}$

STEP 8

Now that we have the slope $m = \frac{1}{2}$ and the y-intercept $b = 0$ , we can write the equation of the function:
$y = \frac{1}{2}x$

STEP 9

To graph the function, we plot the points given in the table and draw a straight line through them.

STEP 10

The points to plot are: (0, 0), (2, 1), (4, 2), (6, 3), and (8, 4).

STEP 11

After plotting these points on a coordinate plane, we draw a line that passes through all of them.

STEP 12

To determine if the relationship is proportional, we need to check if the ratio of $y$ to $x$ is constant for all pairs of $x$ and $y$ .

STEP 13

Calculate the ratio for each pair of $x$ and $y$ from the table:
For $(x, y) = (2, 1)$ , the ratio is $\frac{y}{x} = \frac{1}{2}$ .
For $(x, y) = (4, 2)$ , the ratio is $\frac{y}{x} = \frac{2}{4} = \frac{1}{2}$ .
For $(x, y) = (6, 3)$ , the ratio is $\frac{y}{x} = \frac{3}{6} = \frac{1}{2}$ .
For $(x, y) = (8, 4)$ , the ratio is $\frac{y}{x} = \frac{4}{8} = \frac{1}{2}$ .

STEP 14

Since the ratio $\frac{y}{x}$ is constant and equal to $\frac{1}{2}$ for all pairs of $x$ and $y$ , the relationship is proportional.
The equation for the function shown in the table is $y = \frac{1}{2}x$ . The graph of this function is a straight line through the origin with a slope of $\frac{1}{2}$ . The relationship is proportional because the ratio of $y$ to $x$ is constant for all values in the table.

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Question Two partners split profits evenly. Find the equation, graph, and determine if proportional for $x$ (profit) and $y$ (each partner's share).

Studdy Solution

STEP 1

Assumptions
1. The table provides pairs of $x$ (profit) and $y$ (amount received by each partner).
2. The profits are split evenly between the two business partners.
3. We are to assume that the relationship between $x$ and $y$ can be represented by a function.

STEP 2

To write an equation for the function shown in the table, we need to determine the relationship between $x$ and $y$ . We can observe that for every increase in $x$ , there is a corresponding increase in $y$ .

STEP 3

By examining the table, we can see that when $x$ increases by 2, $y$ increases by 1. This suggests a linear relationship between $x$ and $y$ .

STEP 4

To find the equation of the line, we can use the slope-intercept form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

STEP 5

Since the y-intercept is the value of $y$ when $x = 0$ , we can see from the table that $b = 0$ .

STEP 6

To find the slope $m$ , we can use the change in $y$ over the change in $x$ between any two points from the table.

STEP 7

Using the points (2, 1) and (4, 2), we calculate the slope as follows:
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 1}{4 - 2} = \frac{1}{2}$

STEP 8

Now that we have the slope $m = \frac{1}{2}$ and the y-intercept $b = 0$ , we can write the equation of the function:
$y = \frac{1}{2}x$

STEP 9

To graph the function, we plot the points given in the table and draw a straight line through them.

STEP 10

The points to plot are: (0, 0), (2, 1), (4, 2), (6, 3), and (8, 4).

STEP 11

After plotting these points on a coordinate plane, we draw a line that passes through all of them.

STEP 12

To determine if the relationship is proportional, we need to check if the ratio of $y$ to $x$ is constant for all pairs of $x$ and $y$ .

STEP 13

STEP 14

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Question Two partners split profits evenly. Find the equation, graph, and determine if proportional for xxx (profit) and yyy (each partner's share).

Studdy Solution

STEP 1

Assumptions1. The table provides pairs of xxx (profit) and yyy (amount received by each partner).2. The profits are split evenly between the two business partners.3. We are to assume that the relationship between xxx and yyy can be represented by a function.

STEP 2

To write an equation for the function shown in the table, we need to determine the relationship between xxx and yyy. We can observe that for every increase in xxx, there is a corresponding increase in yyy.

STEP 3

By examining the table, we can see that when xxx increases by 2, yyy increases by 1. This suggests a linear relationship between xxx and yyy.

STEP 4

To find the equation of the line, we can use the slope-intercept form y=mx+by = mx + by=mx+b, where mmm is the slope and bbb is the y-intercept.

STEP 5

Since the y-intercept is the value of yyy when x=0x = 0x=0, we can see from the table that b=0b = 0b=0.

STEP 6

To find the slope mmm, we can use the change in yyy over the change in xxx between any two points from the table.

STEP 7

Using the points (2, 1) and (4, 2), we calculate the slope as follows:m=y2−y1x2−x1=2−14−2=12m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 1}{4 - 2} = \frac{1}{2}m=x2​−x1​y2​−y1​​=4−22−1​=21​

STEP 8

Now that we have the slope m=12m = \frac{1}{2}m=21​ and the y-intercept b=0b = 0b=0, we can write the equation of the function:y=12xy = \frac{1}{2}xy=21​x

STEP 9

To graph the function, we plot the points given in the table and draw a straight line through them.

STEP 10

The points to plot are: (0, 0), (2, 1), (4, 2), (6, 3), and (8, 4).

STEP 11

After plotting these points on a coordinate plane, we draw a line that passes through all of them.

STEP 12

To determine if the relationship is proportional, we need to check if the ratio of yyy to xxx is constant for all pairs of xxx and yyy.

STEP 13

STEP 14

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Question Two partners split profits evenly. Find the equation, graph, and determine if proportional for $x$ (profit) and $y$ (each partner's share).

Assumptions
1. The table provides pairs of $x$ (profit) and $y$ (amount received by each partner).
2. The profits are split evenly between the two business partners.
3. We are to assume that the relationship between $x$ and $y$ can be represented by a function.

To write an equation for the function shown in the table, we need to determine the relationship between $x$ and $y$ . We can observe that for every increase in $x$ , there is a corresponding increase in $y$ .

By examining the table, we can see that when $x$ increases by 2, $y$ increases by 1. This suggests a linear relationship between $x$ and $y$ .

To find the equation of the line, we can use the slope-intercept form $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

Since the y-intercept is the value of $y$ when $x = 0$ , we can see from the table that $b = 0$ .

To find the slope $m$ , we can use the change in $y$ over the change in $x$ between any two points from the table.

Using the points (2, 1) and (4, 2), we calculate the slope as follows:
$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 1}{4 - 2} = \frac{1}{2}$

Now that we have the slope $m = \frac{1}{2}$ and the y-intercept $b = 0$ , we can write the equation of the function:
$y = \frac{1}{2}x$

To determine if the relationship is proportional, we need to check if the ratio of $y$ to $x$ is constant for all pairs of $x$ and $y$ .