Question

How do you find an equation of the line that contains the following pair of points \[\left( { - 4, - 5} \right)\] and \[\left( { - 8, - 10} \right)\] ?

Answer 1

Hint: In this problem we have to find the equation of a line with two points. The equation of a line passing through two points \[\left( {{x_1},{\text{ }}{y_1}} \right)\] and \[\left( {{x_2},{\text{ }}{y_2}} \right)\] is given by \[\dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{x - {x_1}}}{{{x_2} - {x_1}}}\] . Hence, we will use this two-point form of the equation to get the required equation of the line. After that we will rearrange the equation in the standard form i.e., \[ax + by + c = 0\] . and hence we will get the required equation of the line.

Complete step by step answer:
We have given two points on the line.
Let \[\left( { - 4, - 5} \right)\] be \[\left( {{x_1},{\text{ }}{y_1}} \right)\] and \[\left( { - 8, - 10} \right)\] be \[\left( {{x_2},{\text{ }}{y_2}} \right)\]
We have to find the equation of the line passing through these two points.
Now we know that according to the two-point form, the equation of a line passing through two points \[\left( {{x_1},{\text{ }}{y_1}} \right)\] and \[\left( {{x_2},{\text{ }}{y_2}} \right)\] is given by
\[\dfrac{{y - {y_1}}}{{{y_2} - {y_1}}} = \dfrac{{x - {x_1}}}{{{x_2} - {x_1}}}\]
Here, \[{x_1} = - 4,{\text{ }}{x_2} = - 8,{\text{ }}{y_1} = - 5,{\text{ }}{y_2} = - 10\]
So, on substituting the values, we get
\[ \Rightarrow \dfrac{{y - \left( { - 5} \right)}}{{\left( { - 10} \right) - \left( { - 5} \right)}} = \dfrac{{x - \left( { - 4} \right)}}{{\left( { - 8} \right) - \left( { - 4} \right)}}\]
On simplification, we get
\[ \Rightarrow \dfrac{{y + 5}}{{ - 5}} = \dfrac{{x + 4}}{{ - 4}}\]
Now cross multiplying the equation, we get
\[ \Rightarrow - 4\left( {y + 5} \right) = - 5\left( {x + 4} \right)\]
On multiplying, we get
\[ \Rightarrow - 4y - 20 = - 5x - 20\]
\[ \Rightarrow - 4y = - 5x\]
Now rearranging the equation in the standard form i.e., \[ax + by + c = 0\] we get
\[ - 5x + 4y + 0 = 0\]
Taking \[ - 1\] common, we get
\[5x - 4y = 0\]
Hence the required equation passing through \[\left( { - 4, - 5} \right)\] and \[\left( { - 8, - 10} \right)\] is \[5x - 4y = 0\]
We can see the same information in the below graph:

Note:
This question can also be solved using slope intercept form.
We know that the slope intercept form of the line is given by
\[y = mx + c{\text{ }} - - - \left( i \right)\]
where \[m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\] is the slope and \[c\] is the y-intercept.
Now we have two points,
Let \[\left( {{x_1},{\text{ }}{y_1}} \right) = \left( { - 4, - 5} \right)\] and \[\left( {{x_2},{\text{ }}{y_2}} \right) = \left( { - 8, - 10} \right)\]
Therefore, on substituting the values
Slope, \[m = \dfrac{{\left( { - 10} \right) - \left( { - 5} \right)}}{{\left( { - 8} \right) - \left( { - 4} \right)}}\]
\[ \Rightarrow m = \dfrac{{ - 10 + 5}}{{ - 8 + 4}}\]
\[ \Rightarrow m = \dfrac{5}{4}\]
On substituting in equation \[\left( i \right)\] we get
\[y = \dfrac{5}{4}x + c\]
Now substitute the value of \[x\] and \[y\] from any of the two points to get the y-intercept.
Let’s take the point \[\left( { - 4, - 5} \right)\]
Therefore, we have
\[ - 5 = \dfrac{5}{4}\left( { - 4} \right) + c\]
\[ \Rightarrow c = 0\]
Thus, the y-intercept is \[0\] and the slope is \[\dfrac{5}{4}\]
Therefore, slope intercept form of the line will be
\[y = \dfrac{5}{4}x\]
On rearranging in the standard form, we get
\[5x - 4y = 0\]
Hence, we get the required equation of the line passing through \[\left( { - 4, - 5} \right)\] and \[\left( { - 8, - 10} \right)\]