Question

How do you find the equation of a line passing through $\left( 4,1 \right)$ with the slope $m=-\dfrac{1}{2}?$

Answer 1

Hint: The equation of a line is typically written as $y=mx+b$ where $m$ is the slope and $b$ is the $y$ intercept. We can find the equation of a straight line when given the slope and a point on the line by using the formula $y=m\left( x-{{x}_{1}} \right)$ where $m$ is slope and $\left( y-{{y}_{1}} \right)$ is point on the line. After putting value in the formula and solving then we can find the equation.

Complete step-by-step answer:
As per the question we have point $\left( 4,1 \right)$ and the slope $m=-\dfrac{1}{2}$ we have to find the equation of line using the given data.
The equation of a line is typically written as $y=mx+b$ where $m$ is slope and $b$ is $y$ intercept so we will not use this formula.
We will use point slope form for finding line equations.
Point slope formula $y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$
Where $m$ is the slope and $\left( {{x}_{1}},{{y}_{1}} \right)$ is the point.
$\Rightarrow$ $y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)...(i)$
$\Rightarrow$ $m=-\dfrac{1}{2}\left( {{x}_{1}},{{y}_{1}} \right)=\left( 4,1 \right)$
Now put these value in the equation $(i)$
$\Rightarrow$ $y-{{y}_{1}}=m\left( x-{{x}_{1}} \right)$
$\Rightarrow$ $y-1=-\dfrac{1}{2}\left( x-4 \right)$
$\Rightarrow$ $2\left( y-1 \right)=-1\left( x-4 \right)$
$\Rightarrow$ $2y-2=-1x+4$
$\Rightarrow$ $2y=-1x+4+2$
$\Rightarrow$ $2y=-1x+6$
$\Rightarrow$ $y=-\dfrac{1}{2}x+3$
So, the equation of a line passing through $\left( 4,1 \right)$ with the slope $m=-\dfrac{1}{2}$ is
$\Rightarrow$ $y=-\dfrac{1}{2}x+3$
By the point slope form we can find the equation of line.
Additional Information:
We can solve this by slope intercept form. We will solve one example in slope intercept form i.e. $y=mx+b$ where $m$ is the slope $b$ is the $y$ intercept and $x$ and $y$ stay. Written as $x$ and $y$ in the final equation.
Since we already have the slope our equation is now.
$y=-\dfrac{4}{5}x+b$ (because $m$ represents the slope so we plug. The slope’s value inform)
Now, we must find the $y$-intercept. In order to find $y$ intercept we simply use the point given. By putting in $4$ in the place of $x$ and $2$ for $y$
We will get,
$\Rightarrow$ $y=-\dfrac{4}{5}x+b$
$\Rightarrow$ $2=-\dfrac{4}{5}\left( 4 \right)+b$
$\Rightarrow$ $2=\dfrac{16}{5}+b$
$\Rightarrow$ $b=-\dfrac{4}{5}$
So, we put the value of $b$ and the value of $m$ in the equation and we get our final equation
$\Rightarrow$ $y=-\dfrac{4}{5}x-\dfrac{4}{5}$
This method of solving is called a slope-intercept. From this we solve this with various methods.

Note:
First while solving this problem you should know the line intercept formula and write that correctly. If the given data is $\left( 0,2 \right)$ and $\left( 5,2 \right)$ then we will have another method solving various methods of solving this type of problem.