Question

Which of the following pair of linear equations has infinite solutions?

• A. $x+2y=7$; $3x+6y=21$
• B. $4x+3y=7$; $3x+6y=25$
• C. $y+2x=10$; $11x+6y=21$
• D. $\frac{1}{2}x+2y=\frac{7}{11}$; $x+6y=21$

Answer 1

The correct option is A

$x+2y=7$; $3x+6y=21$

If two equations are consistent and overlapping, then they will have infinite solutions. Option A consists of two equations where the second equation can be reduced to an equation which is same as the first equation.
$x+2y=7$ ....(i)
$3x+6y=21$ .....(ii)
Dividing equation (ii) by 3, we get
$x+2y=7$ which is the same as equation (i).
The equations coincide and will have an infinite solution.
Alternate Method:
Let the two equations be
$a_{1}x+b_{1}y+c_1=0$
$a_{2}x+b_{2}y+c_2=0$
The condition for having infinite solutions is:
$\frac{a_1}{a_2}=\frac{b_1}{b_2}=\frac{c_1}{c_2}$ ...(i)
For the equations , on substituting values in eq (i), we get
$\frac{1}{3}=\frac{2}{6}=\frac{-7}{-21}$
$\therefore$ The pair of equations $x+2y=7$ and $3x+6y=21$ . have infinite solutions.
Similarly, we can check that other options don't have infinite solutions.