Question

How do you write the standard form of the equation given $\left( { - 1,2} \right)$ and slope $2$ ?

Answer 1

Hint: Here we will find the straight line equation with slope and one given point by using the slope point form. On doing some simplification we get the required answer.

Formula used: Let $m$ be the slope of a straight line and $\left( {{x_1},{y_1}} \right)$ be a point on a straight line.
Let $\left( {x,y} \right)$be an arbitrary point on a straight line other than $\left( {{x_1},{y_1}} \right)$. Then, we have,
$m = \dfrac{{y - {y_1}}}{{x - {x_1}}} \Leftrightarrow m\left( {x - {x_1}} \right) = y - {y_1}$
Thus, the equation of a straight line with slope $m$ and passing through
 $\left( {{x_1},{y_1}} \right)$is $y - {y_1} = m\left( {x - {x_1}} \right)$---------(1)
For all points $\left( {x,y} \right)$ on straight-line

Complete step-by-step solution:
Here the slope is $m = 2$ and the point on a straight line is $\left( { - 1,2} \right)$
So, ${x_1} = - 1$ and ${y_1} = 2$ then substitute this value into the general equation of a straight line with slope $m$.
That is $y - {y_1} = m\left( {x - {x_1}} \right)$
$ \Rightarrow y - 2 = 2\left( {x - ( - 1)} \right)$
On rewriting we get
$ \Rightarrow y - 2 = 2\left( {x + 1} \right)$
Multiply $2$ into the bracket we get,
$ \Rightarrow y - 2 = 2x + 2$
Reordering the terms like this we get,
$ \Rightarrow 2x - y + 2 + 2 = 0$
Let us add the term and we get
$ \Rightarrow 2x - y + 4 = 0$

Therefore the equation of a straight line is $2x - y + 4 = 0$

Note: The first degree equation (1) in the variables $x$ and $y$ is satisfied by the $x$ coordinate and $y$ coordinate of any point on the line. Any value of $x$ and $y$ that satisfies this equation will be the coordinates of a point on the line. Hence the equation (1) is called the equation of the straight line.
The equation (1) says that the change in $y$ coordinates of the points on a straight line is directly proportional to the change in $x$ coordinates. The proportionality constant $m$ is the slope.
If the line with slope $m$ , makes $x$ intercept $d$ , then the equation of the line is $y = m(x - d)$ .
The straight line $y = mx$ passes through the origin (both $x$ and $y$ intercepts are zero for $m \ne 0$ ).