Question

One equation of a pair of dependent linear equations 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: To obtain the second equation of a pair of dependent linear equations we will check each option and see which one can be written as the equation given. Firstly we will take the value in the equation then we will simplify it by taking a number common among all terms and try to check whether it can be written as the equation given if not move on to other options till we get our desired answer.

Complete step-by-step solution:
One equation of a pair of dependent linear equations is given as:
$-5x+7y=2$……$\left( 1 \right)$
We will start by option (a)
$10x+14y+4=0$
Taking 2 common from each term and write it in the form of equation (1)
$\begin{align}
  & 2\left( 5x+7y+2 \right)=0 \\
 & \Rightarrow 5x+7y+2=0 \\
 & \Rightarrow 5x+7y=-2 \\
\end{align}$
It can’t be written as equation (1) so it is not dependent equation (1).
Next take option (b)
$-10x-14y+4=0$
Taking 2 common from each term and write it in the form of equation (1)
$\begin{align}
  & 2\left( -5x-7y+2 \right)=0 \\
 & \Rightarrow -5x-7y+2=0 \\
 & \Rightarrow -5x-7y=-2 \\
\end{align}$
It can’t be written as equation (1) so it is not dependent equation (1).
Next we will take option (c)
$-10x+14y+4=0$
Taking 2 common from each term and write it in the form of equation (1)
$\begin{align}
  & 2\left( -5x+7y+2 \right)=0 \\
 & \Rightarrow -5x+7y+2=0 \\
 & \Rightarrow -5x+7y=-2 \\
\end{align}$
It can’t be written as equation (1) so it is not dependent equation (1).
Finally take option (d)
$10x-14y=-4$
Taking 2 common from each term and write it in the form of equation (1)
$\begin{align}
  & 2\left( 5x-7y \right)=-4 \\
 & \Rightarrow 5x-7y=-2 \\
 & \Rightarrow -5x+7y=2 \\
\end{align}$
It is the same as equation (1) so it is dependent on equation (1).
Hence correct option is (d).

Note: We can use another method to find the solution that is by using the condition for dependent linear equation where the below value should be satisfied:
$\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}=\dfrac{{{c}_{1}}}{{{c}_{2}}}=\dfrac{1}{k}$
Where, ${{a}_{1}},{{b}_{1}},{{c}_{1}}$ is coefficient of one equation and ${{a}_{2}},{{b}_{2}},{{c}_{2}}$ is the coefficient of another equation and $k$ can be any arbitrary constant.