Question

How do you find the $x{\text{ and }}y$ intercepts of the line containing points $\left( {4, - 2} \right),\left( {5, - 4} \right)$?

Answer 1

Hint: Here we need to proceed by finding the slope of the line given and then we can find the equation of the line which contains these two points. Now to find the $x$ intercept, we will put $y = 0$ in the equation of the line and for $y$ intercept we will put $x = 0$.

Complete step-by-step answer:
Here we are given that we need to find the $x{\text{ and }}y$ intercepts of the line containing points $\left( {4, - 2} \right),\left( {5, - 4} \right)$
So we know that whenever we have two points given through which the line is passing we can easily find the equation of that line by the formula $(y - {y_1}) = m(x - {x_1})$ where $m$ is the slope of the line having these two points and $({x_1},{y_1})$ can be any of the two points of the line.
We know that slope can be calculated by the formula $\dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ where $({x_1},{y_1}),({x_2},{y_2})$ are the two points of the line. Hence we can substitute the value of the two points in this we will get:
$m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \dfrac{{ - 4 - ( - 2)}}{{5 - 4}} = - 4 + 2 = - 2$
Hence we get the value of the slope as $ - 2$ and now we can write the equation of the line by substituting the value of $m$ and $({x_1},{y_1})$ in the equation of the line and we will get:
$(y - {y_1}) = m(x - {x_1})$
$\Rightarrow$ $(y + 2) = - 2(x - 4)$
On simplifying we get:
$
  y + 2 = - 2x + 8 \\
\Rightarrow 2x + y - 6 = 0 - - - - (3) \\
 $
Now we need to get the $x{\text{ and }}y$ intercept which means we need to get the point where this line meets the $x{\text{ and }}y$ axis.
So we know that at the x axis the $y = 0$ hence put $y = 0$ in equation (3) we will get:
$
  2x - 6 = 0 \\
\Rightarrow x = 3 \\
 $
Hence we get x intercept as $3$ and the coordinate as $\left( {3,0} \right)$
Now similarly to get the y intercept we need to put $x = 0$ and we will get:
$
  y - 6 = 0 \\
\Rightarrow y = 6 \\
 $
So we get y intercept as $6$ and the coordinate as $\left( {0,6} \right)$.

Note: Here in these types of problems we need to mainly focus on finding the equation of the line properly. We can also be given the slope of the line with one point on it instead of two points. So we must know the formula to find the equation of the line in any form.