Question

How do you write an equation of a line with point $\left( 2,4 \right)$ and slope 2?

Answer 1

We are given point $\left( 2,4 \right)$ and slope 2, we are asked to find expression that define equation of line, to do so we will learn about slope-point form $\left( y-{{y}_{0}} \right)=m\left( x-{{x}_{0}} \right)$ to get the equation of the line, once we get the line we will convert it to the standard form of line given as $ax+by+c=0$ When we get this form at that point our solution is achieved.

Complete step-by-step solution:
We are given a point $\left( 2,4 \right)$ and the slope 2. We are asked to find the equation of a line.
To find the solution we will understand how to define an equation of line and what slope means with respect to the line.
Now, the slope of any line is the angle made by the line with the positive x-axis.
Slope is given as $\tan \theta$
There is another way to find the slope, we first find two points on the line say $\left( {{x}_{1}},{{y}_{1}} \right)$ and $\left( {{x}_{2}},{{y}_{2}} \right)$ then slope is given as $m=\dfrac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$
Now the standard equation of a line is given as –
$ax+by+c=0$
And there is another form of line which is called point slope form.
Which is given as $y-{{y}_{0}}=m\left( x-{{x}_{0}} \right)$
Where ‘m’ denotes the slope of the line and $\left( {{x}_{0}},{{y}_{0}} \right)$ is our point that lies on the line.
We will use first the point slope form of the line to get the equation of line then we change it into the standard form of line as we have slope of the line as ‘2’, it means we have $m=2$ .
Also we have that the line passes through $\left( 2,4 \right)$ or say $\left( 2,4 \right)$ lies on the line so, we consider $\left( {{x}_{0}},{{y}_{0}} \right)=\left( 2,4 \right)$
Now we have $m=2$ and $\left( {{x}_{2}},{{y}_{0}} \right)=\left( 2,4 \right)$using there in $\left( y-{{y}_{0}} \right)=m\left( x,{{x}_{0}} \right)$ we get –
$y-4=2\left( x-2 \right)$
Opening brackets we get –
$y-4=2x-4$
Now we have our equation as –
$y-4=2x-4$
We will use algebra to change it into standard form we add ‘4’ on both side of $y-4=2x-4$
$y-4+4=2x-4+4$
So, we get –
$y=2x$ $\left( \text{as }-4+4=0 \right)$
Now, subtracting 2x on both side, we get –
$y-2x=2x-2x$
So we get –
$-2x+y=0$ $\left( \text{as }2x-2x=0 \right)$
**Hence, we get our equation of line as –
$-2x+y=0$ in standard form or we simply called $y=2x$ **

Note: While solving the algebraic operation, if we have to multiply a term with the bracket containing multiple terms then we multiply that given term with each entry of the brackets like $a\left( b+c \right)=ab+ac$
If we do it like $a\left( b+c \right)=ab+c$ , then this is an incorrect step, we do not multiply just by one term we solve by multiplying with all terms.