Question Find the y-intercept of the additive relationship between $x$ and $y$ given in the table: $x = \{2, 3, 4, 5, 6\}$ , $y = \{7, 8, 9, 10, 11\}$ .

Studdy Solution

STEP 1

Assumptions
1. The relationship between $x$ and $y$ is additive, which means $y = mx + b$ , where $m$ is the constant rate of change (slope) and $b$ is the $y$ -intercept.
2. The values of $x$ and $y$ are given in the table.
3. We need to find the $y$ -intercept ( $b$ ) of the relationship.

STEP 2

Determine the rate of change (slope) $m$ by calculating the change in $y$ over the change in $x$ between any two points in the table.
$m = \frac{\Delta y}{\Delta x}$

STEP 3

Choose two points from the table to calculate the slope. We can use the first two points $(2, 7)$ and $(3, 8)$ .
$m = \frac{8 - 7}{3 - 2}$

STEP 4

Calculate the slope $m$ .
$m = \frac{1}{1} = 1$

STEP 5

Now that we have the slope, we can use the point-slope form of the equation to find the $y$ -intercept $b$ . We can use any point from the table for this purpose. Let's use the first point $(2, 7)$ .
$y - y_1 = m(x - x_1)$

STEP 6

Substitute the slope $m$ and the coordinates of the point $(x_1, y_1)$ into the point-slope form equation.
$y - 7 = 1(x - 2)$

STEP 7

Simplify the equation to find the $y$ -intercept $b$ .
$y = x - 2 + 7$

STEP 8

Combine like terms to get the equation in the form $y = mx + b$ .
$y = x + 5$

STEP 9

From the equation $y = x + 5$ , we can see that the $y$ -intercept $b$ is 5.
The $y$ -intercept of this additive relationship is 5.

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Question Find the y-intercept of the additive relationship between $x$ and $y$ given in the table: $x = \{2, 3, 4, 5, 6\}$ , $y = \{7, 8, 9, 10, 11\}$ .

Studdy Solution

STEP 1

Assumptions
1. The relationship between $x$ and $y$ is additive, which means $y = mx + b$ , where $m$ is the constant rate of change (slope) and $b$ is the $y$ -intercept.
2. The values of $x$ and $y$ are given in the table.
3. We need to find the $y$ -intercept ( $b$ ) of the relationship.

STEP 2

Determine the rate of change (slope) $m$ by calculating the change in $y$ over the change in $x$ between any two points in the table.
$m = \frac{\Delta y}{\Delta x}$

STEP 3

Choose two points from the table to calculate the slope. We can use the first two points $(2, 7)$ and $(3, 8)$ .
$m = \frac{8 - 7}{3 - 2}$

STEP 4

Calculate the slope $m$ .
$m = \frac{1}{1} = 1$

STEP 5

Now that we have the slope, we can use the point-slope form of the equation to find the $y$ -intercept $b$ . We can use any point from the table for this purpose. Let's use the first point $(2, 7)$ .
$y - y_1 = m(x - x_1)$

STEP 6

Substitute the slope $m$ and the coordinates of the point $(x_1, y_1)$ into the point-slope form equation.
$y - 7 = 1(x - 2)$

STEP 7

Simplify the equation to find the $y$ -intercept $b$ .
$y = x - 2 + 7$

STEP 8

Combine like terms to get the equation in the form $y = mx + b$ .
$y = x + 5$

STEP 9

From the equation $y = x + 5$ , we can see that the $y$ -intercept $b$ is 5.
The $y$ -intercept of this additive relationship is 5.

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Question Find the y-intercept of the additive relationship between xxx and yyy given in the table: x={2,3,4,5,6}x = \{2, 3, 4, 5, 6\}x={2,3,4,5,6}, y={7,8,9,10,11}y = \{7, 8, 9, 10, 11\}y={7,8,9,10,11}.

Studdy Solution

STEP 1

Assumptions1. The relationship between xxx and yyy is additive, which means y=mx+by = mx + by=mx+b, where mmm is the constant rate of change (slope) and bbb is the yyy-intercept.2. The values of xxx and yyy are given in the table.3. We need to find the yyy-intercept (bbb) of the relationship.

STEP 2

Determine the rate of change (slope) mmm by calculating the change in yyy over the change in xxx between any two points in the table.m=ΔyΔxm = \frac{\Delta y}{\Delta x}m=ΔxΔy​

STEP 3

Choose two points from the table to calculate the slope. We can use the first two points (2,7)(2, 7)(2,7) and (3,8)(3, 8)(3,8).m=8−73−2m = \frac{8 - 7}{3 - 2}m=3−28−7​

STEP 4

Calculate the slope mmm.m=11=1m = \frac{1}{1} = 1m=11​=1

STEP 5

Now that we have the slope, we can use the point-slope form of the equation to find the yyy-intercept bbb. We can use any point from the table for this purpose. Let's use the first point (2,7)(2, 7)(2,7).y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​)

STEP 6

Substitute the slope mmm and the coordinates of the point (x1,y1)(x_1, y_1)(x1​,y1​) into the point-slope form equation.y−7=1(x−2)y - 7 = 1(x - 2)y−7=1(x−2)

STEP 7

Simplify the equation to find the yyy-intercept bbb.y=x−2+7y = x - 2 + 7y=x−2+7

STEP 8

Combine like terms to get the equation in the form y=mx+by = mx + by=mx+b.y=x+5y = x + 5y=x+5

STEP 9

From the equation y=x+5y = x + 5y=x+5, we can see that the yyy-intercept bbb is 5.The yyy-intercept of this additive relationship is 5.

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Question Find the y-intercept of the additive relationship between $x$ and $y$ given in the table: $x = \{2, 3, 4, 5, 6\}$ , $y = \{7, 8, 9, 10, 11\}$ .

Assumptions
1. The relationship between $x$ and $y$ is additive, which means $y = mx + b$ , where $m$ is the constant rate of change (slope) and $b$ is the $y$ -intercept.
2. The values of $x$ and $y$ are given in the table.
3. We need to find the $y$ -intercept ( $b$ ) of the relationship.

Determine the rate of change (slope) $m$ by calculating the change in $y$ over the change in $x$ between any two points in the table.
$m = \frac{\Delta y}{\Delta x}$

Choose two points from the table to calculate the slope. We can use the first two points $(2, 7)$ and $(3, 8)$ .
$m = \frac{8 - 7}{3 - 2}$

Calculate the slope $m$ .
$m = \frac{1}{1} = 1$

Now that we have the slope, we can use the point-slope form of the equation to find the $y$ -intercept $b$ . We can use any point from the table for this purpose. Let's use the first point $(2, 7)$ .
$y - y_1 = m(x - x_1)$

Substitute the slope $m$ and the coordinates of the point $(x_1, y_1)$ into the point-slope form equation.
$y - 7 = 1(x - 2)$

Simplify the equation to find the $y$ -intercept $b$ .
$y = x - 2 + 7$

Combine like terms to get the equation in the form $y = mx + b$ .
$y = x + 5$

From the equation $y = x + 5$ , we can see that the $y$ -intercept $b$ is 5.
The $y$ -intercept of this additive relationship is 5.