Question

How do you write the equation in point slope form given $\left( { - 1,10} \right)$ and $\left( {5,8} \right)$ ?

Answer 1

Hint : Here in this question, we have to find the equation of the straight line passing through the two points $\left( {{x_1},{y_1}} \right)$ and $\left( {{x_2},{y_2}} \right)$ . Find the equation by using the Point-Slope formula $y - {y_1} = m\left( {x - {x_1}} \right)$ before finding the equation first we have to find the slope using the formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$ . On simplification to the point-slope formula we get the required solution.

Complete step-by-step answer :
The general equation of a straight line is $y = mx + c$ , where $m$ is the gradient or slope and $\left( {0,c} \right)$ the coordinates of the y-intercept.
Consider, the point-slope formula
$y - {y_1} = m\left( {x - {x_1}} \right)$ -------(1)
The point-slope formula uses the slope and the coordinates of a point along the line to find the y-intercept.
Find the slope $m$ in point-slope formula by using the formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
Where ${x_1} = - 1$ , ${x_2} = 5$ , ${y_1} = 10$ and ${y_2} = 8$ on substituting this in formula, then
$\Rightarrow m = \dfrac{{8 - 10}}{{5 - \left( { - 1} \right)}}$
$\Rightarrow m = \dfrac{{ - 2}}{{5 + 1}}$
$\Rightarrow m = \dfrac{{ - 2}}{6}$
On simplification, we get
$\Rightarrow m = - \dfrac{1}{3}$
Now we get the gradient or slope of the line which passes through the points $\left( { - 1,10} \right)$ and $\left( {5,8} \right)$ .
Substitute the slope m and the point $\left( {{x_1},{y_1}} \right) = \left( { - 1,10} \right)$ in the point slope formula.
Consider the equation (1)
$y - {y_1} = m\left( {x - {x_1}} \right)$
Where $\,m = - \dfrac{1}{3}$ , ${x_1} = - 1$ and ${y_1} = 10$ on substitution, we get
$\Rightarrow y - 10 = \, - \dfrac{1}{3}\left( {x - \left( { - 1} \right)} \right)$
$\Rightarrow y - 10 = \, - \dfrac{1}{3}\left( {x + 1} \right)$
Multiply both side by 3, then
$\Rightarrow 3\left( {y - 10} \right) = \, - \left( {x + 1} \right)$
$\Rightarrow 3y - 30 = \, - x - 1$
Add 30 on both side, then
$\Rightarrow 3y - 30 + 30 = \, - x - 1 + 30$
$\Rightarrow 3y = \, - x + 29$
Divide both side by 3 then
$\Rightarrow y = \,\dfrac{{ - x + 29}}{3}$
or
Or it can be written as
$\Rightarrow y = \, - \dfrac{1}{3}\left( {x - 29} \right)$
Hence, the equation of the line passing through points $\left( { - 1,10} \right)$ and $\left( {5,8} \right)$ is $y = \, - \dfrac{1}{3}\left( {x - 29} \right)$ .
So, the correct answer is “$y = \, - \dfrac{1}{3}\left( {x - 29} \right)$”.

Note : The slope of a line is a ratio of the change in the y value and the change in the x value. We have to know the equation of a line and then we have to substitute the values to the equation, hence we can determine the value. While simplifying the equation we must take care of signs of terms.