Solved on Dec 12, 2023

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

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.

Was this helpful?

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

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.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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}.

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.

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

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.