Question

Write the equation of the line that satisfies the given conditions in standard form: Contains the point (3,4) and is perpendicular to the line 2x - 4y = -3.

Answer 1

The general equation of a line is given by:

`\begin{gathered} y=mx+c \\ \text{where m = slope} \\ c=\text{intercept} \end{gathered}`

Given the line 2x -4y = -3

Step 1: Re-write the equation by making y the subject

`\begin{gathered} 4y=2x+3 \\ y=(2)/(4)x+(3)/(4) \\ y=(1)/(2)x+(3)/(4) \end{gathered}`

Since we have the equation of the line to be

`y=(1)/(2)x+(3)/(4)`

Step 2: Obtain the slope of this line

`\text{slope}=m=(1)/(2)`

Step 3: Obtain the slope perpendicular to this line.

If two line are perpendicular, then

`m_1m_2=-1`

so that

`m_2=-(1)/(m_1)`

`m_2=-(1)/((1)/(2))=-2`

Hence the slope of the new line will be = -2

Step 4: Obtain the equation of the line using the formula:

`y-y_1=m(x-x_1)`

where

`\begin{gathered} x_1=3,y_1=4,and\text{ } \\ m=-2 \end{gathered}`

Thus,

`y-4=-2(x-3)`

=>

`\begin{gathered} y-4=-2x+6 \\ y=-2x+6+4 \\ y=-2x+10 \end{gathered}`

The equation of the line that s perpendicular to the line is

=> y= -2x +10

This can also be written in the form ax+by=c

as

Making the constant to be on the right-hand side and the variables to be on the left and side as shown below

Hence,

2x+y=10