Question

How do you write an equation of the line that passed through point $\left( { - 2,3} \right)$ and is parallel to the line formed by the equation $y = 4x + 7$

Answer 1

Hint : In order to determine the required equation of line, first compare the given equation $y = 4x + 7$ with the general slope form $y = mx + c$ to get the value for slope $m$ . Use the Point-slope form $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$ and put the value of slope $m$ and point $\left( { - 2,3} \right)$ as $\left( {{x_1},{y_1}} \right)$ . By simplifying and the equation , you will get the required equation.

Complete step-by-step answer :
We are given an equation of line as $y = 4x + 7$ and a point $\left( { - 2,3} \right)$ .
In this question we are supposed to find out the equation of line which is parallel to the equation of line $y = 4x + 7$ and passing through the point $\left( { - 2,3} \right)$ .
For this we have to first determine the slope of the line $y = 4x + 7$ .To do so , rewrite the equation of line into the general equation as $y = mx + c$ where $m$ is the slope of the line.
The $y = 4x + 7$ is already in the general form. So, directly compare the equation with the general equation $y = mx + c$ ,we have the slope of line $y = 4x + 7$ equal to $m = 4$ .
Thus, the slope of line parallel to line $y = 4x + 7$ will also have its slope as $m = 4$ .
The Point-Slope Formula of straight line is
$\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$ where $\left( {{x_1},{y_1}} \right)$ is the point on the line.
So, we have the slope of the required line as $m = 4$ and also it is passing through the point $\left( { - 2,3} \right)$ .We can write the equation of straight line using the point slope form as
$\Rightarrow \left( {y - 3} \right) = 4\left( {x - \left( { - 2} \right)} \right) \\\ \Rightarrow \left( {y - 3} \right) = 4\left( {x + 2} \right) \;$
Expanding the bracket on RHS, we get
$\Rightarrow y - 3 = 4x + 8$
combining all the like terms and rewrite the equation into the general equation form as $y = mx + c$ , we can obtain the above equation as
$\Rightarrow y = 4x + 8 + 3 \\\ \Rightarrow y = 4x + 11 \;$
Therefore, the equation of line parallel to $y = 4x + 7$ and passing through the point $\left( { - 2,3} \right)$ is equal to $y = 4x + 11$ .
So, the correct answer is “Option C”.

Note : 1. The graph of both the lines is shown below.
The red line is the graph of equation $y = 4x + 7$ and
The blue line is graph of equation $y = 4x + 11$

You can verify the result as the both the lines are parallel to each other.
2.Slope of line perpendicular to the line having slope $m$ is equal to $- \dfrac{1}{m}$ .
3.We should have a better knowledge in the topic of geometry to solve this type of question easily. We should know the Point-slope form $\left( {y - {y_1}} \right) = m\left( {x - {x_1}} \right)$ where $\left( {{x_1},{y_1}} \right)$ is the point on the line $m$ as the form and also the Slope-intercept form of line as $y = mx + c$ where $m$ is the slope of the line.
4. The general equation for lines parallel to $y = 4x + 7$ will be $y = 4x \pm k$ where $k$ can be any integer.