Question

Find the equation of the line parallel to the line $3x+6y=5$ and passing through the point (1, 3). Write an equation in the slope-intercept form.

Answer 1

To solve this question, we need to know the property of parallel lines. So, we will first change the given equation of line into its slope-intercept form, that is, $y=mx+b$ and then find the value of the slope, that is m. We will then proceed by finding the value of y intercept of the line at (1, 3) and then find the equation of line in the slope intercept form to get the required answer.

Complete step-by-step answer:
We have been given an equation of line $3x+6y=5$, and we have to find the equation of the line parallel to it and passing through the point (1, 3). So, to solve this question, we should know that the parallel lines will have the same slope. The equation of the line given to us is,
$3x+6y=5$
And we have to find the line, parallel to it passing through point (1, 3). Let us take the equation of the line in the slope intercept form as,
$y=mx+b\ldots \ldots \ldots \left( i \right)$
So, we will first convert the equation of the given line in slope intercept form. So, we will get,
$\begin{aligned} & 3x+6y=5 \\\ & \Rightarrow 6y=-3x+5 \\\ & \Rightarrow y=\dfrac{-3}{6}x+\dfrac{5}{6} \\\ & \Rightarrow y=\dfrac{-1}{2}x+\dfrac{5}{6} \\\ \end{aligned}$
Thus we get the slope of the given line as $\dfrac{-1}{2}$= slope of the line as in equation (i), since parallel lines have the same slope.

Now, we will use the slope to find the y-intercept. Now, we know that the slope of the line is $\dfrac{-1}{2}$. So, we will substitute it $m=\dfrac{-1}{2}$ in equation (i). So, we will get,
$y=\dfrac{-1}{2}x+b$
Now, we have been given that the line passes through the point (1, 3). So, using it, we get,
$\begin{aligned} & 3=\dfrac{-1}{2}\times 1+b \\\ & \Rightarrow 3+\dfrac{1}{2}=b \\\ & \Rightarrow b=\dfrac{6+1}{2} \\\ & \Rightarrow b=\dfrac{7}{2} \\\ \end{aligned}$
Therefore, we get the equation of the line parallel to the given line, $3x+6y=5$ and passing through the point (1, 3) as, $y=\dfrac{-1}{2}x+\dfrac{7}{2}$.
So the equation of line is $x+2y=7$
Note: The y intercept for the line $3x+6y=5$ is $\dfrac{5}{6}$. The y intercept has nothing to do with the final answer to the given question and can be ignored. If the lines were perpendicular, then the slope of the two lines must be negative reciprocals of each other, that is ${{m}_{1}}.{{m}_{2}}=-1\Rightarrow {{m}_{1}}=\dfrac{-1}{{{m}_{2}}}\text{ and }{{m}_{2}}=\dfrac{-1}{{{m}_{1}}}$.