Question

Which of the following represents a pair of dependent equations?

• A. $y=-\frac{3}{2}-\frac{x}{2};x=-3-2y$
• B. $5x=6-3y;9y=12-15x$
• C. $y=\frac{2}{7}-\frac{x}{7}$; $x=\frac{2}{7}-\frac{y}{7}$
• D. $3x=6-3y;10y=12-15x$

Answer 1

The correct option is A

$y=-\frac{3}{2}-\frac{x}{2};x=-3-2y$

Consider a pair of linear equations
$a_{1}x+b_{1}y+c_1=0$ and $a_{2}x+b_{2}y+c_2=0$
The equations are said to be dependent if $\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ ...(i)

Consider the following equations
$y=-\frac{3}{2}-\frac{x}{2};x=-3-2y$
By writing them in the standard form, we get
$\frac{x}{2}+y+\frac{3}{2}=0$
$x+2y+3=0$

Here, $a_1=\frac{1}{2},b_1=1and{c}_1=\frac{3}{2}$
Similarly, $a_2=1,b_2=2and{c}_2=3$

On substituting these values in eq. (i), we get
$\frac{\frac{1}{2}}{1}=\frac{1}{2}=\frac{\frac{3}{2}}{3}$

Thus, the condition for dependent lines is satisfied.
Hence,the equations $y=-\frac{3}{2}-\frac{x}{2};x=-3-2y$ are dependent.