Question

How do you find the slope of a line parallel to $x = 4y + 7$?

Answer 1

In this question, we need to find the slope of a line which is parallel to the given line. Firstly, we will convert the given equation into a form of slope intercept form of a straight line. It can be done by first subtracting 7 from both sides of the given equation. Then dividing each term by 4. We then compare the given equation of a line with the standard slope intercept form of a line and find the slope. Then we use the result that two straight lines are parallel if they have the same slope.

Complete step by step solution:
Given an equation of a straight line $x = 4y + 7$ …… (1)
We are asked to find the slope of a straight line which is parallel to the given line.
So firstly we will try to find out the slope of a line given in the equation (1).
To find this, we need to convert our given equation into slope intercept form of a straight line.
The general equation of a straight line in slope intercept form is given by,
$y = mx + c$ …… (2)
where $m$ is the slope or gradient of a line and $c$ is the intercept of a line.
Now we convert the given equation of a line into slop intercept form by rearranging the terms.
Consider the equation of a line given in the equation (1).
This equation can also be written as,
$4y + 7 = x$
Now subtract 7 from both the sides we get,
$\Rightarrow 4y + 7 - 7 = x - 7$
$\Rightarrow 4y + 0 = x - 7$
$\Rightarrow 4y = x - 7$
Now divide throughout by 4 we get,
$\Rightarrow \dfrac{{4y}}{4} = \dfrac{{x - 7}}{4}$
$\Rightarrow y = \dfrac{1}{4}x - \dfrac{7}{4}$
Now comparing with the slope intercept form given in the equation (2), we get,
$m = \dfrac{1}{4}$ and $c = - \dfrac{7}{4}$.
Therefore, the slope of the straight line $x = 4y + 7$ is $m = \dfrac{1}{4}$.
We have the result that if any two lines are parallel, then they have the same slope.

Hence, the slope of a line parallel to $x = 4y + 7$ is $\dfrac{1}{4}$.

Note: It is important to know the general slope intercept form of a straight line which is given by $y = mx + c$, where $m$ is the slope and $c$ is the y- intercept.
Remember that the two straight lines are parallel if they have the same slope and different intercept. Also remember to convert the given equation of a line into a standard slope intercept form. Then only we can get the required value for the slope.
A straight line is perpendicular to x-axis and parallel to y-axis if its equation is of the form $x = c$.
A straight line is perpendicular to y-axis and parallel to x-axis if its equation is of the form $y = c$.