Question

How do you write the equation in slope intercept form given $( - 5,0)$ and $(3,3)$?

Answer 1

In this question, we need to find the equation of a line in slope intercept form. Firstly, we need to find the slope of the given line using the given points. To find the slope we use the formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$. After finding the slope of the line, we substitute in the equation of a line given by $y - {y_1} = m(x - {x_1})$ and find the equation of a given line.

Complete step by step answer:
Here we are given the two points $( - 5,0)$ and $(3,3)$.
We are asked to find the equation of a line which passes through the above given points.
Now we represent the given two points as,
$({x_1},{y_1}) = ( - 5,0)$
$({x_2},{y_2}) = (3,3)$
Firstly, we need to find out the slope of the given line.
The slope of any given line is found using the formula, $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
where $m$ represents the slope of the line.
Now substituting the values of the variable in the above formula we get,
$m = \dfrac{{3 - 0}}{{3 - ( - 5)}}$
$\Rightarrow m = \dfrac{3}{{3 + 5}}$
$\Rightarrow m = \dfrac{3}{8}$
Now we have found the value of the slope of a given line.
We place this value in the equation of a line to obtain the required equation of a line formed by the two points.
The formula to find the equation of a line is given by,
$y - {y_1} = m(x - {x_1})$
where $x,$ $y$ are constants and ${y_1},$ ${x_1}$are variables.
Substituting the values we get,
$\Rightarrow y - 0 = \dfrac{3}{8}(x - ( - 5))$
Solving it further we get,
$\Rightarrow y = \dfrac{3}{8}(x + 5)$
$\Rightarrow y = \dfrac{3}{8}x + \dfrac{3}{8}(5)$
$\Rightarrow y = \dfrac{3}{8}x + \dfrac{{15}}{8}$
Now this is in the form of $y = mx + c$ which is a standard slope intercept form.
Hence comparing the above obtained equation with the standard slope intercept form we get,
$m = \dfrac{3}{8}$ and $c = \dfrac{{15}}{8}$ which are the slope and y-intercept of the given line.

Hence, the equation of a line with the given points $( - 5,0)$ and $(3,3)$ in slope intercept form is given by $y = \dfrac{3}{8}x + \dfrac{{15}}{8}$.

Note: The slope of a line is the steepness of a line in a horizontal or vertical direction. The slope of a line can be calculated by taking the ratio of the change in vertical dimensions upon the change in horizontal dimensions.
i.e. slope, $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Also remember the formula to find the equation of a line given by $y - {y_1} = m(x - {x_1})$ and substitute the given values.
The standard slope intercept form of a straight line is given by $y = mx + c$, where $m$ is the slope of the line and $c$ is the y-intercept