# What is the equation of the line passing through (34,5) and (4,-31)?

Nov 11, 2015

$y = \frac{6 x - 179}{5}$.

#### Explanation:

We will set up the co-ordinates as:
$\left(34 , 5\right)$
$\left(4 , - 31\right)$.

Now we do subtraction of the $x$s and the $y$s.

$34 - 4 = 30$,
$5 - \left(- 31\right) = 36$.

We now divide the difference in $y$ over that in $x$.

$\frac{36}{30} = \frac{6}{5}$.
So $m$ (gradient) $= \frac{6}{5}$.

Equation of a straight line:
$y = m x + c$. So, let's find $c$. We substitute values of any of the coordinates and of $m$:
$5 = \frac{6}{5} \cdot 34 + c$,
$5 = \frac{204}{5} + c$,
$c = 5 - \frac{204}{5}$,
$c = - \frac{179}{5}$. So,

$y = \frac{6 x - 179}{5}$.

Nov 11, 2015

$\textcolor{b l u e}{y = \frac{6}{5} x - 35.8}$

#### Explanation:

Standard form equation is:

$\textcolor{b l u e}{y = m x + c \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(1\right)}$

Where m is the slope (gradient) and c is the point where the plot crosses the y-axis in this context.

The gradient is the amount of up (or down) of y for the amount of along for the x-axis. $\textcolor{b l u e}{\text{Always considered from left to right .}}$

So $m \to \frac{{y}_{2} - {y}_{1}}{{x}_{2} - {x}_{1}} = \frac{\left(- 31\right) - 5}{4 - 34}$

As $\left(34 , 5\right)$ is listed first you assume this is the left most point of the two.

$m = \frac{- 36}{- 30}$ dividing negative into negative gives positive

$\textcolor{b l u e}{m = \frac{36}{30} = \frac{6}{5} \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(2\right)}$

Substitute (2) into (1) giving:

$\textcolor{b l u e}{y = \frac{6}{5} x + c \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \left(3\right)}$

Now all we need to do is substitute known values for x and y to obtain that for c

Let $\left(x , y\right) \to \left(34 , 5\right)$

Then $y = \frac{6}{5} x + c \text{ }$ becomes:

$\textcolor{b r o w n}{5 = \left(\frac{6}{5} \times 34\right) + c}$ $\textcolor{w h i t e}{\times x}$brackets used for grouping only

Subtract $\textcolor{g r e e n}{\left(\frac{6}{5} \times 34\right)}$ from both sides giving

$\textcolor{b r o w n}{5} - \textcolor{g r e e n}{\left(\frac{6}{5} \times 34\right)} \textcolor{w h i t e}{\times} = \textcolor{w h i t e}{\times} \textcolor{b r o w n}{\left(\frac{6}{5} \times 34\right)} - \textcolor{g r e e n}{\left(\frac{6}{5} \times 34\right)} \textcolor{b r o w n}{+ c}$

$c = 5 - \left(\frac{6}{5} \times 34\right)$

$\textcolor{b l u e}{c = - 35.8 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \left(4\right)}$

Substitute (4) into (3) giving:

$\textcolor{b l u e}{y = \frac{6}{5} x - 35.8}$