# How do you write the equation of line given x-intercept of 4 and a slope of 3/4?

Jun 30, 2016

$y = \frac{3}{4} x - 3$

#### Explanation:

The equation of a line in $\textcolor{b l u e}{\text{slope-intercept form}}$ is

$\textcolor{red}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{y = m x + b} \textcolor{w h i t e}{\frac{a}{a}} |}}}$
where m represents the slope and b, the y-intercept.

Here we have a coordinate point (4 ,0) and $m = \frac{3}{4}$

So the partial equation is $y = \frac{3}{4} x + b$ and to find b we substitute x = 4 and y = 0 into the partial equation.

$\Rightarrow 0 = \frac{3}{\cancel{4}} \times \cancel{4} + b = 3 + b \Rightarrow b = - 3$

$\Rightarrow y = \frac{3}{4} x - 3 \text{ is the equation}$

Jun 30, 2016

$y = \frac{3}{4} x - 3$

#### Explanation:

Let $\textcolor{p u r p \le}{k}$ be some constant such that
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} x + \textcolor{p u r p \le}{k}$
where $\textcolor{g r e e n}{m}$ is the slope.

We are told that $\textcolor{g r e e n}{m} = \textcolor{g r e e n}{\frac{3}{4}}$
and
the x-intercept is $\textcolor{b r o w n}{4}$

If the x-intercept is $\textcolor{b r o w n}{4}$
this means that $x = \textcolor{b r o w n}{4}$ when $y = \textcolor{c y a n}{0}$

So our general equation: $y = \textcolor{g r e e n}{m} x + \textcolor{p u r p \le}{k}$
becomes
$\textcolor{w h i t e}{\text{XXX}} \textcolor{c y a n}{0} = \textcolor{g r e e n}{\frac{3}{4}} \cdot \textcolor{b r o w n}{4} + \textcolor{p u r p \le}{k}$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow \textcolor{p u r p \le}{k} = \textcolor{b l u e}{- 3}$

and our equation is
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{\frac{3}{4}} x \textcolor{b l u e}{- 3}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Using the above logic we can see that the general case when given
a slope of $\textcolor{g r e e n}{m}$ and
an x-intercept of $\textcolor{p u r p \le}{k}$
is
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} x \textcolor{b l u e}{- \frac{k}{m}}$