# How do you write the equation in point slope form given (3,5) and (8,15)?

Apr 13, 2018

$y = 2 x - 1$

#### Explanation:

First, we have to solve for the slope using this equation:

$m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

This formula makes sense because slope is rise over run. Rise being the $y$ values and run being the $x$ values.

Now you choose which point is point 2 (includes ${y}_{2}$ and ${x}_{2}$) and which point is point 1 (includes ${y}_{1}$ and ${x}_{1}$)

Point 2: $\left(3 , 5\right) \to {y}_{2} = 5$ and ${x}_{2} = 3$
Point 1: $\left(8 , 15\right) \to {y}_{1} = 15$ and ${x}_{1} = 8$

Plug into the formula and solve for the slope:

$m = \frac{5 - 15}{3 - 8}$

$m = \frac{- 10}{- 5}$

$m = \frac{10}{5}$

$m = 2$

Now we know that the slope, $m = 2$

Next, we have to use point-slope formula to get the equation:

$y - {y}_{1} = m \left(x - {x}_{1}\right)$

You can plug in either point for the ${x}_{1}$ and ${y}_{1}$ values. Let's use the point $\left(3 , 5\right)$.

$y - 5 = m \left(x - 3\right)$

Now plug in the slope

$y - 5 = 2 \left(x - 3\right)$

Distribute the $2$

$y - 5 = 2 x - 6$

Add $5$ to both sides

$y - 5 + 5 = 2 x - 6 + 5$

$y = 2 x - 1$

$y = 2 x - 1$

Apr 13, 2018

color(indigo)(y - 5 = 2*(x - 3) " is the point - slope form of equation"

#### Explanation:

If two points on a line are known, we can use the following formula to write the equation :

$\frac{y - {y}_{1}}{{y}_{2} - {y}_{1}} = \frac{x - {x}_{1}}{{x}_{2} - {x}_{1}}$

"Given : (x_1, y_1) = (3,5), (x_2,y_2) = (8,15)

Hence, equation of the line is

$\frac{y - 5}{15 - 5} = \frac{x - 3}{8 - 3}$

$\frac{y - 5}{\cancel{10}} ^ \textcolor{red}{2} = \frac{x - 3}{\cancel{5}}$

color(purple)(y - 5 = 2*(x - 3)

Standard form of Point-Slope equation is

$y - {y}_{1} = m \cdot \left(x - {x}_{1}\right)$

Apr 13, 2018

Equation of the line in Slope-Intercept Form: color(green)(y=2x-1

#### Explanation:

If the slope $\left(m\right)$ of a line and the coordinate $\left({x}_{1} , {y}_{1}\right)$ of one end-point of the line is known, we can write the equation of the line in Point-Slope Form:

color(red)(y-y_1=m(x-x_1)

To find the slope if we are given two end-points of a line, we use the formula:

color(blue)(Slope(m)=(y_2-y_1)/(x_2-x_1)

We are given the points: $\left(3 , 5\right) \mathmr{and} \left(8 , 15\right)$

Note that x_1 = 3; y_1=5; x_2=8 and y_2=15

$S l o p e \left(m\right) = \frac{15 - 5}{8 - 3}$

$\Rightarrow m = \frac{10}{5}$

$\Rightarrow S l o p e \left(m\right) = 2$

Next, consider the equation of the point-slope form.

Consider the point $\left(3 , 5\right)$

We also found that $S l o p e \left(m\right) = 2$

${x}_{1} = 3 \mathmr{and} {y}_{1} = 5$

$y - {y}_{1} = m \left(x - {x}_{1}\right)$

$\Rightarrow y - 5 = 2 \left(x - 3\right)$

$\Rightarrow y - 5 = 2 x - 6$

Add $5$ to both sides of the equation.

$\Rightarrow y - 5 + 5 = 2 x - 6 + 5$

$\Rightarrow y - \cancel{5} + \cancel{5} = 2 x - 6 + 5$

rArr color(green)(y=2x-1

This is the required equation in the Point-Slope Form.

We can also graph the line and find the intercepts.

Substitute $y = 0$ to obtain the x-intercept.

$2 x - 1 = 0$

Add $1$ to both sides.

$\Rightarrow 2 x - 1 + 1 = 0 + 1$

$\Rightarrow 2 x - \cancel{1} + \cancel{1} = 0 + 1$

$\Rightarrow 2 x = 1$

Divide both sides by $2$

$\frac{2 x}{2} = \frac{1}{2}$

$\frac{\cancel{2} x}{\cancel{2}} = \frac{1}{2}$

$x = \frac{1}{2}$

$x = 0.5$

Hence $\left(0.5 , 0\right)$ is the x-intercept.

Substitute $x = 0$ to obtain the y-intercept.

$y = 2 \left(0\right) - 1$

$y = 0 - 1$

$y = - 1$

Hence $\left(0 , - 1\right)$ is the y-intercept.