Question

How do you find the equation of line parallel to the line parallel to $3x + 5y = 11$ and has a y-intercept of -6?

Answer 1

Hint: Since they have given us the y-intercept and a line try to find the slope using slope intercept form. Now since in this case we have been given y-intercept use the slope found earlier and the y-intercept given in the slope intercept form and you get the equation a line fulfilling the given conditions.

Complete step by step solution:
In this question we have been given a line $3x + 5y = 11$ and asked to find a line parallel to the given line and passing through the given y-intercept.
Let us start the solution by finding the slope of the line given to us.
The slope of the line can be calculated by using the slope intercept form which is $y = {\text{m}}x + {\text{b}}$.
So solving our given equation so that it comes in slope intercept form we get
$
  \Rightarrow 3x + 5y = 11 \\
   \Rightarrow 5y = 11 - 3x \\
   \Rightarrow y = \dfrac{{11}}{5} - \dfrac{3}{5}x \\
   \Rightarrow y = - \dfrac{3}{5}x + \dfrac{{11}}{5} \\
 $
Now comparing this equation with the slope intercept form we get our ${\text{m = }}\dfrac{{ - 3}}{5}$
Therefore the slope of the line $3x + 5y = 11$ is ${\text{m = }}\dfrac{{ - 3}}{5}$. Since parallel lines have the same slope, the line which we are about to find will have the same slope that is ${\text{m = }}\dfrac{{ - 3}}{5}$.
Now since we have been given the y-intercept which is -6, substituting the value of our y-intercept and slope in the slope intercept form we get
$
  \Rightarrow y = - \dfrac{3}{5}x + ( - 6) \\
   \Rightarrow y = - \dfrac{3}{5}x - 6 \\
 $
Solving further,
Multiplying by 5 throughout the equation we get
$
  \Rightarrow 5y = - 3x - 30 \\
   \Rightarrow 3x + 5y = - 30 \\
 $

Hence $3x + 5y = - 30$is the equation of the line which is parallel to the line parallel to $3x + 5y = 11$ and has a y-intercept of -6.

Note: In this question we have been given the y-intercept directly. But in some questions if the y-intercept is not given and instead a interval is given, we should substitute the values of the interval in the slope intercept form to obtain the value of b which is our y-intercept and then proceed with the regular step.