Question Find the equation of the line passing through the points $(6, 20)$ and $(8, 42)$ .

Studdy Solution

STEP 1

Assumptions
1. The graph of the function is a straight line.
2. The line passes through the points $(6,20)$ and $(8,42)$ .
3. The equation of a line in slope-intercept form is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

STEP 2

To find the equation of the line, we first need to calculate the slope ( $m$ ) of the line using the two given points. The slope is the change in $y$ divided by the change in $x$ .
$m = \frac{y_2 - y_1}{x_2 - x_1}$

STEP 3

Now, plug in the coordinates of the given points into the slope formula.
$m = \frac{42 - 20}{8 - 6}$

STEP 4

Calculate the slope.
$m = \frac{22}{2}$

STEP 5

Simplify the slope.
$m = 11$

STEP 6

With the slope known, we can use one of the points and the slope to find the y-intercept ( $b$ ) of the line. We can use the point-slope form of the equation of a line, which is $y - y_1 = m(x - x_1)$ .

STEP 7

Choose one of the given points to plug into the point-slope form. We'll use the point $(6,20)$ .
$20 - y_1 = 11(6 - x_1)$

STEP 8

Now, plug in the values for $x_1$ and $y_1$ from the chosen point.
$20 - 20 = 11(6 - 6)$

STEP 9

Simplify the equation.
$0 = 11(0)$

STEP 10

Since the left side of the equation is $0$ , we can see that the y-intercept $b$ is equal to the $y$ value of the point we chose, which is $20$ .
$b = 20$

STEP 11

Now that we have both the slope ( $m = 11$ ) and the y-intercept ( $b = 20$ ), we can write the equation of the line in slope-intercept form.
$y = mx + b$

STEP 12

Plug in the values for $m$ and $b$ into the slope-intercept form equation.
$y = 11x + 20$
The equation of the function is $y = 11x + 20$ .

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Question Find the equation of the line passing through the points $(6, 20)$ and $(8, 42)$ .

Studdy Solution

STEP 1

Assumptions
1. The graph of the function is a straight line.
2. The line passes through the points $(6,20)$ and $(8,42)$ .
3. The equation of a line in slope-intercept form is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

STEP 2

To find the equation of the line, we first need to calculate the slope ( $m$ ) of the line using the two given points. The slope is the change in $y$ divided by the change in $x$ .
$m = \frac{y_2 - y_1}{x_2 - x_1}$

STEP 3

Now, plug in the coordinates of the given points into the slope formula.
$m = \frac{42 - 20}{8 - 6}$

STEP 4

Calculate the slope.
$m = \frac{22}{2}$

STEP 5

Simplify the slope.
$m = 11$

STEP 6

With the slope known, we can use one of the points and the slope to find the y-intercept ( $b$ ) of the line. We can use the point-slope form of the equation of a line, which is $y - y_1 = m(x - x_1)$ .

STEP 7

Choose one of the given points to plug into the point-slope form. We'll use the point $(6,20)$ .
$20 - y_1 = 11(6 - x_1)$

STEP 8

Now, plug in the values for $x_1$ and $y_1$ from the chosen point.
$20 - 20 = 11(6 - 6)$

STEP 9

Simplify the equation.
$0 = 11(0)$

STEP 10

Since the left side of the equation is $0$ , we can see that the y-intercept $b$ is equal to the $y$ value of the point we chose, which is $20$ .
$b = 20$

STEP 11

Now that we have both the slope ( $m = 11$ ) and the y-intercept ( $b = 20$ ), we can write the equation of the line in slope-intercept form.
$y = mx + b$

STEP 12

Plug in the values for $m$ and $b$ into the slope-intercept form equation.
$y = 11x + 20$
The equation of the function is $y = 11x + 20$ .

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Question Find the equation of the line passing through the points (6,20)(6, 20)(6,20) and (8,42)(8, 42)(8,42).

Studdy Solution

STEP 1

Assumptions1. The graph of the function is a straight line.2. The line passes through the points (6,20)(6,20)(6,20) and (8,42)(8,42)(8,42).3. The equation of a line in slope-intercept form is y=mx+by = mx + by=mx+b, where mmm is the slope and bbb is the y-intercept.

STEP 2

To find the equation of the line, we first need to calculate the slope (mmm) of the line using the two given points. The slope is the change in yyy divided by the change in xxx.m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}m=x2​−x1​y2​−y1​​

STEP 3

Now, plug in the coordinates of the given points into the slope formula.m=42−208−6m = \frac{42 - 20}{8 - 6}m=8−642−20​

STEP 4

Calculate the slope.m=222m = \frac{22}{2}m=222​

STEP 5

Simplify the slope.m=11m = 11m=11

STEP 6

With the slope known, we can use one of the points and the slope to find the y-intercept (bbb) of the line. We can use the point-slope form of the equation of a line, which is y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​).

STEP 7

Choose one of the given points to plug into the point-slope form. We'll use the point (6,20)(6,20)(6,20).20−y1=11(6−x1)20 - y_1 = 11(6 - x_1)20−y1​=11(6−x1​)

STEP 8

Now, plug in the values for x1x_1x1​ and y1y_1y1​ from the chosen point.20−20=11(6−6)20 - 20 = 11(6 - 6)20−20=11(6−6)

STEP 9

Simplify the equation.0=11(0)0 = 11(0)0=11(0)

STEP 10

Since the left side of the equation is 000, we can see that the y-intercept bbb is equal to the yyy value of the point we chose, which is 202020.b=20b = 20b=20

STEP 11

Now that we have both the slope (m=11m = 11m=11) and the y-intercept (b=20b = 20b=20), we can write the equation of the line in slope-intercept form.y=mx+by = mx + by=mx+b

STEP 12

Plug in the values for mmm and bbb into the slope-intercept form equation.y=11x+20y = 11x + 20y=11x+20The equation of the function is y=11x+20y = 11x + 20y=11x+20.

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Question Find the equation of the line passing through the points $(6, 20)$ and $(8, 42)$ .

Assumptions
1. The graph of the function is a straight line.
2. The line passes through the points $(6,20)$ and $(8,42)$ .
3. The equation of a line in slope-intercept form is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

To find the equation of the line, we first need to calculate the slope ( $m$ ) of the line using the two given points. The slope is the change in $y$ divided by the change in $x$ .
$m = \frac{y_2 - y_1}{x_2 - x_1}$

Now, plug in the coordinates of the given points into the slope formula.
$m = \frac{42 - 20}{8 - 6}$

Calculate the slope.
$m = \frac{22}{2}$

Simplify the slope.
$m = 11$

With the slope known, we can use one of the points and the slope to find the y-intercept ( $b$ ) of the line. We can use the point-slope form of the equation of a line, which is $y - y_1 = m(x - x_1)$ .

Choose one of the given points to plug into the point-slope form. We'll use the point $(6,20)$ .
$20 - y_1 = 11(6 - x_1)$

Now, plug in the values for $x_1$ and $y_1$ from the chosen point.
$20 - 20 = 11(6 - 6)$

Simplify the equation.
$0 = 11(0)$

Since the left side of the equation is $0$ , we can see that the y-intercept $b$ is equal to the $y$ value of the point we chose, which is $20$ .
$b = 20$

Now that we have both the slope ( $m = 11$ ) and the y-intercept ( $b = 20$ ), we can write the equation of the line in slope-intercept form.
$y = mx + b$

Plug in the values for $m$ and $b$ into the slope-intercept form equation.
$y = 11x + 20$
The equation of the function is $y = 11x + 20$ .