Question

How do you write an equation in standard form given the point $\left( {1, - 1} \right)$and $(3,5)$?

Answer 1

In the given problem, we are required to find the equation of a line when two points lying on the straight line are given to us. We can easily find the equation of the line using the two points form of a straight line. So, we have to substitute the coordinates of both the points given to us in the slope and point form of the line and then simplify the equation of the straight line required.

Complete step by step solution:
So, we are given that points $\left( {1, - 1} \right)$ and $\left( {3,5} \right)$ lie on the line.
We know the two point form of the line, where we can find the equation of a straight line given any two points lying on the line. The slope point form of the line can be represented as: $\dfrac{{\left( {y - {y_1}} \right)}}{{\left( {x - {x_1}} \right)}} = \dfrac{{\left( {{y_2} - {y_1}} \right)}}{{\left( {{x_2} - {x_1}} \right)}}$ where $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ are the points lying on the line given to us.
Considering ${x_1} = 1$ and ${y_1} = - 1$ as the first point given to us is $(1, - 1)$.
Also, ${x_2} = 3$ and ${y_2} = 5$ as the first point given to us is $(3,5)$.
Therefore, required equation of line is as follows:
$\dfrac{{\left( {y - \left( { - 1} \right)} \right)}}{{\left( {x - \left( 1 \right)} \right)}} = \dfrac{{\left( {\left( 5 \right) - \left( { - 1} \right)} \right)}}{{\left( {3 - 1} \right)}}$
On opening the brackets and simplifying further, we get,
$\Rightarrow \dfrac{{\left( {y + 1} \right)}}{{\left( {x - 1} \right)}} = \dfrac{{\left( {5 + 1} \right)}}{{\left( {3 - 1} \right)}}$
Doing the calculations,
$\Rightarrow \dfrac{{y + 1}}{{x - 1}} = \dfrac{6}{2}$
Cancelling the common factors in numerator and denominator, we get,
$\Rightarrow y + 1 = 3\left( {x - 1} \right)$
$\Rightarrow y + 1 = 3x - 3$
$\Rightarrow y = 3x - 4$
Writing the equation of straight line in standard form, we get,
$\Rightarrow y - 3x + 4 = 0$

Hence, the equation of the straight line is: $y - 3x + 4 = 0$.

Note: The given problem requires us to have thorough knowledge of the concepts of coordinate geometry. The equation of the straight line in all the forms must be remembered. The applications of concepts learnt in coordinate geometry are wide ranging.