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.