To determine the equation, first find the slope:

\(\displaystyle \frac{\Delta y}{\Delta x} = \frac{3-2}{2--1} = \frac{1 }{3 }\)

We want this equation in slope-intercept form, \(\displaystyle y = mx+b\). We know \(\displaystyle x\) and \(\displaystyle y\) because we have two coordinate pairs to choose from representing an \(\displaystyle x\) and a \(\displaystyle y\). We know \(\displaystyle m\) because that represents the slope. We just need to solve for \(\displaystyle b\), and then we can write the equation.

We can choose either point and get the correct answer. Let's choose \(\displaystyle (3,2)\): 

\(\displaystyle 2= \frac{1}{3} (3) + b\) multiply "\(\displaystyle mx\)"

\(\displaystyle 2 = 1 + b\) subtract \(\displaystyle 1\) from both sides

\(\displaystyle 1 = b\)

This means that the \(\displaystyle y = mx+b\) form is \(\displaystyle y = \frac{1}{3}x +1\)

To determine the equation, first find the slope:

\(\displaystyle \frac{\Delta y}{\Delta x} = \frac{-7--1}{3-5} = \frac{-6 }{-2 } =3\)

We want this equation in slope-intercept form, \(\displaystyle y = mx+b\). We know \(\displaystyle x\) and \(\displaystyle y\) because we have two coordinate pairs to choose from representing an \(\displaystyle x\) and a \(\displaystyle y\) . We know \(\displaystyle m\) because that represents the slope. We just need to solve for \(\displaystyle b\), and then we can write the equation.

We can choose either point and get the correct answer. Let's choose \(\displaystyle (3, -7)\): 

\(\displaystyle -7 = 3 (3) + b\) multiply "\(\displaystyle mx\)"

\(\displaystyle -7 =9+ b\) subtract \(\displaystyle 6\) from both sides

\(\displaystyle -16 = b\)

This means that the \(\displaystyle y = mx+b\) form is \(\displaystyle y = 3x -16\)

To determine the equation, first find the slope:

\(\displaystyle \frac{\Delta y}{\Delta x} = \frac{2--4}{6--3} = \frac{6}{9 } =\frac{2}{3}\)

We want this equation in slope-intercept form, \(\displaystyle y = mx+b\). We know \(\displaystyle x\) and \(\displaystyle y\) because we have two coordinate pairs to choose from representing an \(\displaystyle x\) and a \(\displaystyle y\). We know \(\displaystyle m\) because that represents the slope. We just need to solve for \(\displaystyle b\), and then we can write the equation.

We can choose either point and get the correct answer. Let's choose \(\displaystyle (6, 2)\): 

\(\displaystyle 2 = \frac{2}{3} (6) + b\) multiply "\(\displaystyle mx\)"

\(\displaystyle 2 = 4 + b\) subtract \(\displaystyle 4\) from both sides

\(\displaystyle -2 = b\)

This means that the \(\displaystyle y = mx+b\) form is \(\displaystyle y = \frac{2}{3}x-2\)

First, determine the slope of the line using the slope formula:

\(\displaystyle \frac{ \Delta y }{ \Delta x } = \frac{ y_2 - y_1 }{ x _ 2 - x_1 } = \frac{7 - 3} { 5 - 2 } = \frac{4}{3}\)

The equation will be in the form \(\displaystyle y = mx + b\) where m is the slope that we just determined, and b is the y-intercept. To determine that, we can plug in the slope for m and the coordinates of one of the original points for x and y:

\(\displaystyle 3 = \frac{4}{3} (2) + b\)

\(\displaystyle 3 = \frac{8 }{3 } + b\) to subtract, it will be easier to convert 3 to a fraction, \(\displaystyle \frac{9}{3}\)

\(\displaystyle \frac{9}{3} - \frac{8}{3} = b\)

\(\displaystyle \frac{1}{3} = b\)

The equation is \(\displaystyle y = \frac{4}{3} x + \frac{1}{3}\)

First, find the slope of the line:

\(\displaystyle \frac{ \Delta y }{ \Delta x } = \frac{2 - 1 }{ 4 - - 2 } = \frac{ 1 }{ 6 }\)

Now we want to find the y-intercept. We can figure this out by plugging in the slope for "m" and one of the points in for x and y in the formula \(\displaystyle y = mx+b\):

\(\displaystyle 1 = \frac{1}{6} (-2 ) + b\)

\(\displaystyle 1 = -\frac{1}{3} + b\)

\(\displaystyle 1 \frac{1}{3} = b\)

The equation is \(\displaystyle y = \frac{ 1}{6} x + 1 \frac{1}{3}\)

To find the equation of a line passing through these points we must find a line with that same slope. Start by finding the slope between the two points and then use the point slope equation to find the equation of the line.

slope:

\(\displaystyle m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{0--5}{5-0}=1\)

Now use the point slope equation:

\(\displaystyle y-y_{1}=m(x-x_{1})\rightarrow y-(-5)=1(x-0)\rightarrow \mathbf{y=x-5}\)

*make sure you use the SAME coordinate pair when substituting x and y into the point slope equation.

Recall that the slope-intercept form of a line:

\(\displaystyle y=mx+b\),

where \(\displaystyle m=\text{slope}\) and \(\displaystyle b=\text{y-intercept}\).

First, find the slope of the line by using the following formula:

\(\displaystyle \text{Slope of Line}=\frac{y_2-y_1}{x_2-x_1}\)

\(\displaystyle \text{Slope}=\frac{5-12}{-2-3}=\frac{7}{5}\)

Next, find the y-intercept of the line by plugging in \(\displaystyle 1\) of the points into the semi-completed formula.

\(\displaystyle y=\frac{7}{5}x+b\)

Plugging in \(\displaystyle (3, 12)\) yields the following:

\(\displaystyle 12=\frac{7}{5}(3)+b\)

Solve for \(\displaystyle b\).

\(\displaystyle b+\frac{21}{5}=12\)

\(\displaystyle b=\frac{39}{5}\)

The equation of the line is then \(\displaystyle y=\frac{7}{5}x+\frac{39}{5}\).

Recall that the slope-intercept form of a line:

\(\displaystyle y=mx+b\),

where \(\displaystyle m=\text{slope}\) and \(\displaystyle b=\text{y-intercept}\).

First, find the slope of the line by using the following formula:

\(\displaystyle \text{Slope of Line}=\frac{y_2-y_1}{x_2-x_1}\)

\(\displaystyle \text{Slope}=\frac{8-7}{12-(-10)}=\frac{1}{22}\)

Next, find the y-intercept of the line by plugging in \(\displaystyle 1\) of the points into the semi-completed formula.

\(\displaystyle y=\frac{1}{22}x+b\)

Plugging in \(\displaystyle (12, 8)\) yields the following:

\(\displaystyle 8=\frac{1}{22}(12)+b\)

Solve for \(\displaystyle b\).

\(\displaystyle b+\frac{6}{11}=8\)

\(\displaystyle b=\frac{82}{11}\)

The equation of the line is then \(\displaystyle y=\frac{1}{22}x+\frac{82}{11}\).

Recall that the slope-intercept form of a line:

\(\displaystyle y=mx+b\),

where \(\displaystyle m=\text{slope}\) and \(\displaystyle b=\text{y-intercept}\).

First, find the slope of the line by using the following formula:

\(\displaystyle \text{Slope of Line}=\frac{y_2-y_1}{x_2-x_1}\)

\(\displaystyle \text{Slope}=\frac{10-2}{16-15}=8\)

Next, find the y-intercept of the line by plugging in \(\displaystyle 1\) of the points into the semi-completed formula.

\(\displaystyle y=8x+b\)

Plugging in \(\displaystyle (16, 10)\) yields the following:

\(\displaystyle 10=8(16)+b\)

Solve for \(\displaystyle b\).

\(\displaystyle b+128=10\)

\(\displaystyle b=-118\)

The equation of the line is then \(\displaystyle y=8x-118\).

Recall that the slope-intercept form of a line:

\(\displaystyle y=mx+b\),

where \(\displaystyle m=\text{slope}\) and \(\displaystyle b=\text{y-intercept}\).

First, find the slope of the line by using the following formula:

\(\displaystyle \text{Slope of Line}=\frac{y_2-y_1}{x_2-x_1}\)

\(\displaystyle \text{Slope}=\frac{8-(-5)}{-4-(-3)}=-13\)

Next, find the y-intercept of the line by plugging in \(\displaystyle 1\) of the points into the semi-completed formula.

\(\displaystyle y=-13x+b\)

Plugging in \(\displaystyle (-3, -5)\) yields the following:

\(\displaystyle -5=-13(-3)+b\)

Solve for \(\displaystyle b\).

\(\displaystyle b+39=-5\)

\(\displaystyle b=-44\)

The equation of the line is then \(\displaystyle y=-13x-44\).