# What is the equation of the line passing through (3,7) and (-8,12)?

Nov 20, 2015

It is important that you be more specific in your question. I am assuming you mean a 'strait line graph'.

$\textcolor{g r e e n}{y = - \frac{5}{11} x + \frac{92}{11}}$
You did not request the y and x intercept so not calculated

#### Explanation:

$\textcolor{g r e e n}{\text{The standard (std) form of this type of plot is}}$

$\textcolor{g r e e n}{y = m x + c}$

color(blue)(~~~~~~~~~~~~ "Pre-amble"~~~~~~~~~~~~~~~~~)

$\textcolor{b r o w n}{y \text{ is the dependent variable as it is the outcome of and thus}}$
$\textcolor{b r o w n}{\text{controlled by what is on the right of the =}}$

$\textcolor{b r o w n}{x \text{ is the independent variable as it can take on any value you}}$
$\textcolor{b r o w n}{\text{chose.}}$

$\textcolor{b r o w n}{m \textcolor{w h i t e}{x} \text{is the gradient of the 'curve'. Yes it is mathematically correct}}$
$\textcolor{b r o w n}{\text{to call a strait line plot a curve. People do not tend to}}$
$\textcolor{b r o w n}{\text{though!}}$

$\textcolor{b l u e}{\approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx \approx}$

$\textcolor{g r e e n}{\text{We start by determining the gradient. We then}}$
$\textcolor{g r e e n}{\text{substitute a pair of the given co-ordinates (Ordered pairs)}}$
$\textcolor{g r e e n}{\text{to find the value of the constant.}}$

$\textcolor{b l u e}{\text{Find the gradient}}$

$m = \left(\text{change in up or down")/("change in along}\right) \to \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}}$

Let $\left({x}_{1} , {y}_{1}\right) = \left(3 , 7\right)$
the left most pair chosen to be so as you listed them first!

Let $\left({x}_{2} , {y}_{2}\right) = \left(- 8 , 12\right)$

Then $m = \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{12 - 7}{\left(- 8\right) - 3} = \frac{5}{- 11}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Find the constant}}$

Using ; $y = m x + c$

then for$\textcolor{w h i t e}{\times} \left({x}_{1} , {y}_{1}\right) \to {y}_{1} = m {x}_{1} + c$

$\implies 7 = \left(- \frac{5}{11}\right) 3 + c$

$c = 7 + \frac{15}{11} = 8 \frac{4}{11} \text{ or } \frac{92}{11}$
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Putting it all together!}}$

Thus, we now have what we need: the gradient and the constant

The equation is : $y = - \frac{5}{11} x + \frac{92}{11}$

$\textcolor{g r e e n}{\text{Fractions are precise:::: Decimals are less so!!! Use fractions in}}$
$\textcolor{g r e e n}{\text{preference unless specifically instructed otherwise!!!!!!!}}$ 