Question

How do you write an equation of a line that passes through $\left( { - 5,9} \right)$ and $\left( {1,3} \right)$?

Answer 1

To find the equation of a line first we have to find the slope and this can be done by dividing the difference of $y$-coordinates of 2 end-points on a line by the difference of $x$-coordinates of the same endpoints. The slope of the line can be a positive or negative value, and the formula is given by, Slope of the line$m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$, and the equation of line that passes through the points is given by,
$\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$, here, ${x_1}$ and ${x_2}$ are $x$-coordinates and ${y_1}$ and ${y_2}$ are $y$ coordinates on $x$-axis and $y$-axis respectively, now substituting the values in the formula we will get the required equation.

Complete step by step solution:
The slope of the line is used to calculate the steepness of the line, it is usually denoted by the letter ‘$m$’. It is the change in$y$divide by the change in$x$, and it is given by the formula,
Slope of the line$m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$.
Here given points are $\left( { - 5,9} \right)$ and $\left( {1,3} \right)$,
So by the formula, here ${x_1} = - 5$, ${y_1} = 9$,${x_2} = 1$,${y_2} = 3$,
Now substituting the values in the formula we get,
Slope of line $m = \dfrac{{3 - 9}}{{1 - \left( { - 5} \right)}}$,
Now simplifying we get,
Slope $m = \dfrac{{ - 6}}{{1 + 5}}$,
Now again simplifying we get,
Slope $m = \dfrac{{ - 6}}{6}$,
Now again simplifying the fraction we get,
Slope $m = - 1$,
Now slope $m = - 1$.
And the equation of line that passes through the points is given by,
$\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$,
So, here ${x_1} = - 5$,${y_1} = 9$ and $m = - 1$,
By substituting the values in the equation we get,
$\Rightarrow y - 9 = \left( { - 1} \right)\left( {x - \left( { - 5} \right)} \right)$,
Now simplifying we get,
$\Rightarrow y - 9 = \left( { - 1} \right)\left( {x + 5} \right)$,
Now multiplying we get,
$\Rightarrow y - 9 = - x - 5$,
Now take all terms to one side we get,
$\Rightarrow x + 5 + y - 9 = 0$,
Now simplifying we get,
$\Rightarrow x + y - 4 = 0$.
So, the required equation is $x + y - 4 = 0$,

$\therefore$The equation of a line that passes through $\left( { - 5,9} \right)$ and $\left( {1,3} \right)$ will be equal to $x + y - 4 = 0$.

Note:
Remember that if the slope of a line is equal to zero then it is parallel to x-axis and if the slope tends to infinity then it is perpendicular to x-axis. Also, we can remember that if the x-coordinates of the two points through which the line passes are same it must be perpendicular to the x-axis and y-coordinates of the two points through which the line passes are same it must be parallel to the x-axis.