Solved on Dec 17, 2023

Find the equation of the line passing through the points (1,6) and (2,7).

STEP 1

Assumptions
1. The line passes through the points $(1,6)$ and $(2,7)$ .
2. 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

First, we need to find the slope of the line that passes through the two given points. The slope $m$ is defined as 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 two points into the slope formula.
Let $(x_1, y_1) = (1, 6)$ and $(x_2, y_2) = (2, 7)$ .
$m = \frac{7 - 6}{2 - 1}$

STEP 4

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

STEP 5

Now that we have the slope of the line, 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 line equation:
$y - y_1 = m(x - x_1)$

STEP 6

Plug in the slope $m$ and the coordinates of one of the points into the point-slope form. We'll use the point $(1, 6)$ .
$y - 6 = 1(x - 1)$

STEP 7

Now, solve for $y$ to get the equation in slope-intercept form.
$y - 6 = x - 1$

STEP 8

Add 6 to both sides of the equation to isolate $y$ .
$y = x - 1 + 6$

STEP 9

Simplify the equation.
$y = x + 5$

STEP 10

The equation of the line in slope-intercept form is:
$y = x + 5$
This is the equation of the line that passes through the points $(1,6)$ and $(2,7)$ .

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Find the equation of the line passing through the points (1,6) and (2,7).

STEP 1

Assumptions
1. The line passes through the points $(1,6)$ and $(2,7)$ .
2. 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

First, we need to find the slope of the line that passes through the two given points. The slope $m$ is defined as 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 two points into the slope formula.
Let $(x_1, y_1) = (1, 6)$ and $(x_2, y_2) = (2, 7)$ .
$m = \frac{7 - 6}{2 - 1}$

STEP 4

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

STEP 5

Now that we have the slope of the line, 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 line equation:
$y - y_1 = m(x - x_1)$

STEP 6

Plug in the slope $m$ and the coordinates of one of the points into the point-slope form. We'll use the point $(1, 6)$ .
$y - 6 = 1(x - 1)$

STEP 7

Now, solve for $y$ to get the equation in slope-intercept form.
$y - 6 = x - 1$

STEP 8

Add 6 to both sides of the equation to isolate $y$ .
$y = x - 1 + 6$

STEP 9

Simplify the equation.
$y = x + 5$

STEP 10

The equation of the line in slope-intercept form is:
$y = x + 5$
This is the equation of the line that passes through the points $(1,6)$ and $(2,7)$ .

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Find the equation of the line passing through the points (1,6) and (2,7).

STEP 1

Assumptions1. The line passes through the points (1,6)(1,6)(1,6) and (2,7)(2,7)(2,7).2. 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

First, we need to find the slope of the line that passes through the two given points. The slope mmm is defined as 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 two points into the slope formula.Let (x1,y1)=(1,6)(x_1, y_1) = (1, 6)(x1​,y1​)=(1,6) and (x2,y2)=(2,7)(x_2, y_2) = (2, 7)(x2​,y2​)=(2,7).m=7−62−1m = \frac{7 - 6}{2 - 1}m=2−17−6​

STEP 4

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

STEP 5

Now that we have the slope of the line, 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 line equation:y−y1=m(x−x1)y - y_1 = m(x - x_1)y−y1​=m(x−x1​)

STEP 6

Plug in the slope mmm and the coordinates of one of the points into the point-slope form. We'll use the point (1,6)(1, 6)(1,6).y−6=1(x−1)y - 6 = 1(x - 1)y−6=1(x−1)

STEP 7

Now, solve for yyy to get the equation in slope-intercept form.y−6=x−1y - 6 = x - 1y−6=x−1

STEP 8

Add 6 to both sides of the equation to isolate yyy.y=x−1+6y = x - 1 + 6y=x−1+6

STEP 9

Simplify the equation.y=x+5y = x + 5y=x+5

STEP 10

The equation of the line in slope-intercept form is:y=x+5y = x + 5y=x+5This is the equation of the line that passes through the points (1,6)(1,6)(1,6) and (2,7)(2,7)(2,7).

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Assumptions
1. The line passes through the points $(1,6)$ and $(2,7)$ .
2. The equation of a line in slope-intercept form is $y = mx + b$ , where $m$ is the slope and $b$ is the y-intercept.

First, we need to find the slope of the line that passes through the two given points. The slope $m$ is defined as 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 two points into the slope formula.
Let $(x_1, y_1) = (1, 6)$ and $(x_2, y_2) = (2, 7)$ .
$m = \frac{7 - 6}{2 - 1}$

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

Now that we have the slope of the line, 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 line equation:
$y - y_1 = m(x - x_1)$

Plug in the slope $m$ and the coordinates of one of the points into the point-slope form. We'll use the point $(1, 6)$ .
$y - 6 = 1(x - 1)$

Now, solve for $y$ to get the equation in slope-intercept form.
$y - 6 = x - 1$

Add 6 to both sides of the equation to isolate $y$ .
$y = x - 1 + 6$

Simplify the equation.
$y = x + 5$

The equation of the line in slope-intercept form is:
$y = x + 5$
This is the equation of the line that passes through the points $(1,6)$ and $(2,7)$ .