Question

How do you write an equation for the line parallel to that contains $P\left( {2,7} \right)$?

Answer 1

Given is a linear equation in two variables $x$ and $y$ . It is given in the standard form of slope intercept form, which is given by $y = mx + c$, where the value of the slope is $m$, and the value of the y-intercept is equal to $c$. Parallel lines have the same slope. Using this information we try to solve the problem.

Complete step-by-step solution:
Here the slope of the given linear equation $y = - 8x - 6$ is :
$\Rightarrow m = - 8$
Any two parallel lines have the same slope, so the line parallel to the given line $y = - 8x - 6$, is going to have the same slope.
Now let us assume the equation of the new line parallel to the given line is given by:
$\Rightarrow y = mx + {c_1}$
Where the value of the slope is the same as the given line $y = - 8x - 6$, so the value of the slope is -8.
So the equation of the new parallel line becomes:
$\Rightarrow y = - 8x + {c_1}$
Now given that this line passes through the point $P\left( {2,7} \right)$, now substituting this point in the above equation to get the value of ${c_1}$, as shown below:
$\Rightarrow 7 = - 8\left( 2 \right) + {c_1}$
$\Rightarrow 7 = - 16 + {c_1}$
On further simplification of the above equation, the value of the new intercept becomes:
$\Rightarrow {c_1} = 23$
Now substituting this in the equation $y = - 8x + {c_1}$, as shown below:
$\Rightarrow y = - 8x + 23$
Equation of the line parallel to $y = - 8x - 6$ passing through $P\left( {2,7} \right)$ is $y = - 8x + 23$.

Note: Please note that we found the parallel line to the given line only when we are given a point passing through it. The most important information is that parallel lines have the same slope, whereas for the perpendicular lines the product of their slopes is equal to -1.