# How do you find a standard form equation for the line with (1,3) and (5,9)?

Dec 21, 2017

$y = \frac{3}{2} x + \frac{3}{2}$

#### Explanation:

We must use our knowledge of line graphs:

The general equation is: $y = m x + c$

Where $m$ is our gradient...

To find $m$:

$m = \text{change in y" /"change in x} = \frac{\Delta y}{\Delta x}$

$\implies m = \frac{9 - 3}{5 - 1} = \frac{6}{4} = \frac{3}{2}$

$\implies y = \frac{3}{2} x + c$

Now to find $c$:

We can just substitute one of the points in:

$3 = \left(\frac{3}{2} \cdot 1\right) + c$

Subtracting $\frac{3}{2}$ from both sides...

$3 - \frac{3}{2} = c$

$c = \frac{3}{2}$

$\implies y = \frac{3}{2} x + \frac{3}{2}$

$\implies y = \frac{3}{2} \left(x + 1\right)$ if you understand factorisation...

Dec 21, 2017

$3 x - 2 y = - 3$

#### Explanation:

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

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{A x + B y = C} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{where A is a positive integer and B, C are integers}$

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

•color(white)(x)y=mx+b

$\text{where m is the slope and b the y-intercept}$

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

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{let "(x_1,y_1)=(1,3)" and } \left({x}_{2} , {y}_{2}\right) = \left(5.9\right)$

$\Rightarrow m = \frac{9 - 3}{5 - 1} = \frac{6}{4} = \frac{3}{2}$

$\Rightarrow y = \frac{3}{2} x + b \leftarrow \textcolor{b l u e}{\text{is the partial equation}}$

$\text{to find b use either of the two given points}$

$\text{substituting "(1,3)" into the partial equation}$

$3 = \frac{3}{2} + b \Rightarrow b = \frac{3}{2}$

$\Rightarrow y = \frac{3}{2} x + \frac{3}{2} \leftarrow \textcolor{red}{\text{in slope-intercept form}}$

$\text{multiply through by 2}$

$2 y = 3 x + 3$

$\Rightarrow 3 x - 2 y = - 3 \leftarrow \textcolor{red}{\text{in standard form}}$