Question

One equation of a pair of dependent linear equation is $ - 5x + 7y = 2$. The second equation can be:
A. $10x + 14y + 4 = 0$
B. $ - 10x - 14y + 4 = 0$
C. $ - 10x + 14y + 4 = 0$
D. $10x - 14y = - 4$

Answer 1

Hint: An equation is said to be linear when it makes a straight line when it is graphed. This type of equation of a straight-line can be written in the form of $y = mx + c$. In the given question we have to check it by the given options one by one as we know that any dependent equation gives an independent equation when it is divided by 2 or we can say that any independent equation gives a dependent equation when it is multiplied by 2. So in the question, we have two procedures to solve it whether we can check it by given options or we can directly multiply it by 2 and get a dependant equation. So first we will check it with options then directly multiply it by 2 to get the correct answer.

Complete step by step answer:

Given that:
$ - 5x + 7y = 2.........\left( 1 \right)$
We have to find the dependent equation of above equation
We will check through the options:
Option A
$10x + 14y + 4 = 0$
On dividing by 2
We get
$
  5x + 7y + 2 = 0 \\
   \Rightarrow 5x + 7y = - 2 \\
 $
It cannot be written exactly as equation 1, so it is not a dependent equation of 1
Now from option B
$ - 10x - 14y + 4 = 0$
$
   \Rightarrow - 5x - 7y + 2 = 0 \\
   \Rightarrow 5x + 7y = 2 \\
 $
It cannot be written exactly as equation$1$, so it is not a dependent equation of $1$

Now from option C
$ - 10x + 14y + 4 = 0$
$
   \Rightarrow - 5x + 7y + 2 = 0 \\
   \Rightarrow - 5x + 7y = - 2 \\
 $
It cannot be written exactly as equation$1$, so it is not a dependent equation of $1$
Now from option D
$10x - 14y = - 4$
$
   \Rightarrow 5x - 7y = - 2 \\
   \Rightarrow - 5x + 7y = 2 \\
 $
It is the same as equation $1$, so it is the dependent equation of $1$
Hence the correct answer to this problem is option D.
OR
We can directly multiply the independent equation by 2 and get a dependant equation
The given equation is:
$ - 5x + 7y = 2$
Multiplied it by 2
We get
$
  \left( { - 5x + 7y = 2} \right) \times 2 \\
   \Rightarrow - 10x + 14y = 4 \\
   \Rightarrow 10x - 14y - 4 \\
 $
Hence the correct answer of this problem is option D.

Note: In the given question we have to find the dependent equation for the given equation. For this, we can use two methods. First is multiply the independent equation by two and we get the second dependent equation for it or we can check it by options, as we did in the solution by dividing the given options by $2$. Hence we get the correct answer.