# How do you write an equation in point slope form that passes through (1,4) and(4,9)?

Jun 3, 2018

$y = \textcolor{g r e e n}{\frac{5}{3}} x + \textcolor{p u r p \le}{\frac{7}{3}}$

#### Explanation:

We want this in the form $y = \textcolor{g r e e n}{m} x + \textcolor{p u r p \le}{c}$, where $m$ is the gradient, and $c$ is the y-intercept.

Gradient is the change in y over the change in x, or the rise over run; however you like to think about it.

$\textcolor{g r e e n}{m} = \frac{\textcolor{red}{{y}_{2}} - \textcolor{b l u e}{{y}_{1}}}{\textcolor{red}{{x}_{2}} - \textcolor{b l u e}{{x}_{1}}}$

So for our points $\left(\textcolor{b l u e}{1 , 4}\right)$ and $\left(\textcolor{red}{4 , 9}\right)$

$m = \frac{\textcolor{red}{9} - \textcolor{b l u e}{4}}{\textcolor{red}{4} - \textcolor{b l u e}{1}}$

$= \frac{5}{3}$

Now, by rearranging our equation for gradient, we can show that:

$y - \textcolor{b l u e}{{y}_{1}} = \textcolor{g r e e n}{m} \left(x - \textcolor{b l u e}{{x}_{1}}\right)$

Here, $x$ and $y$ are general points on the line - we've just replaced $\left(\textcolor{red}{{x}_{2} , {y}_{2}}\right)$ with $\left(\textcolor{red}{x , y}\right)$

Since we know $\textcolor{g r e e n}{m}$, and have a set of co-ordinates to sub in as color(blue)((x_1, y_1), we can sub values in, then re-arrange to get to the $y = \textcolor{g r e e n}{m} x + \textcolor{p u r p \le}{c}$ form.

$y - \textcolor{b l u e}{4} = \textcolor{g r e e n}{\frac{5}{3}} \left(x - \textcolor{b l u e}{1}\right)$

$y - 4 = \frac{5}{3} x - \frac{5}{3}$
$y = \frac{5}{3} x - \frac{5}{3} + 4$
$y = \textcolor{g r e e n}{\frac{5}{3}} x + \textcolor{p u r p \le}{\frac{7}{3}}$
as required.

An alternative method, after finding the gradient, is to substitute your gradient and a pair of points into $y = \textcolor{g r e e n}{m} x + \textcolor{p u r p \le}{c}$, rearrange to find $c$, then go back afterwards and write the equation. If you've been taught that way, its fine! But personally I prefer substituting straight into $y - \textcolor{b l u e}{{y}_{1}} = \textcolor{g r e e n}{m} \left(x - \textcolor{b l u e}{{x}_{1}}\right)$ since you don't have to go back afterwards. Up to you which method you use.

Jun 3, 2018

$y - 4 = \frac{5}{3} \left(x - 1\right)$

#### Explanation:

$\text{the equation of a line in "color(blue)"point-slope form}$ is.

•color(white)(x)y-y_1=m(x-x_1)

$\text{where m is the slope and "(x_1,y_1)" a point on the line}$

$\text{to calculate m use the "color(blue)"gradient formula}$

•color(white)(x)m=(y_2-y_1)/(x_2-x_1)

$\text{let "(x_1,y_1)=(1,4)" and } \left({x}_{2} , {y}_{2}\right) = \left(4 , 9\right)$

$m = \frac{9 - 4}{4 - 1} = \frac{5}{3}$

$\text{use either of the 2 given points as point on line}$

$\text{using "(1,4)" then}$

$y - 4 = \frac{5}{3} \left(x - 1\right) \leftarrow \textcolor{red}{\text{in point-slope form}}$