Question

What is the line equation that has \[x-\] intercept \[=4\] and \[y\]\[-\] intercept \[=-5\]?

Answer 1

Hint: In order to find the line equation, firstly we will be considering the general line equation i.e. \[y=mx+c\] and then we will be determining the value of \[c\] and from that we will be determining the value of \[m\] which is the slope. And upon solving, we will be solving the obtained equations and we will get the equation of line required.

Complete step by step solution:
Now let us learn about the line equations. The general line equation is of the form \[y=mx+c\] where, \[m\] is the slope and \[c\] is the \[y\]\[-\] intercept. There are three major forms of line equations. They are: point-slope form, standard from and slope-intercept form.
Now let us start finding out the line equation that has \[x-\] intercept \[=4\] and \[y\]\[-\] intercept \[=-5\].
Let us consider the equation \[y=mx+c\].
Now we will be determining the value of \[c\].
As the \[x-axis\] crosses the \[y-axis\] at \[x=0\], we will be substituting \[0\] for \[x\].
We get,
\[y=m\left( 0 \right)+c\]
Upon solving, we get, \[y=c\].
From the given question, we have that \[y\]\[-\]intercept\[=-5\]. Since \[y=c\], we can state that \[c=-5\].
Now consider \[y=mx+c\]
Then we get, \[y=mx-5\]
Now, let us determine the value of \[m\] i.e. slope.
\[m=\dfrac{\text{change in y}}{\text{change in x}}\]
Now let us determine the intercepts into the points.
For \[y\]\[-\] intercept, we have \[{{P}_{1}}\left( {{x}_{1}},{{y}_{1}} \right)\] i.e. \[\left( 0,-5 \right)\].
For \[x-\] intercept, we have \[{{P}_{2}}\left( {{x}_{2}},{{y}_{2}} \right)\] i.e. \[\left( 4,0 \right)\].
\[\Rightarrow \]\[m=\dfrac{\text{change in y}}{\text{change in x}}\]\[=\dfrac{{{y}_{2}}-{{y}_{2}}}{{{x}_{2}}-{{x}_{1}}}\]\[=\dfrac{0-\left( -5 \right)}{4-0}=\dfrac{0+5}{4}=\dfrac{5}{4}\]
Now upon substituting the value of \[m\] in the equation \[y=mx-5\], we get the line equation as \[y=\dfrac{5}{4}x-5\].
This can also be simplified and written as \[4y=5x-20\]
\[\therefore \] The line equation that has \[x-\]intercept\[=4\] and \[y\]\[-\]intercept\[=-5\] is \[4y=5x-20\].

Note: For a non vertical line, if it passes through \[\left( x_0,y_0 \right)\] with the slope \[m\], then the equation of line would be \[y-y_0=m\left( x-x_0 \right)\].
Now let us plot our obtained line equation.