Question

Find the values of a and b for which rach of the following systems of linear equations has an infinite number of equations:

$2x+3y=7,2ax+(a+b)y=28.$

Answer 1

The above equations can be rewritten as:

$2x+3y=7,2ax+(a+b)y=28.$

We know that, for the given system of equations to have infinitely many solutions, $\frac{a_1}{a_2}$ = $\frac{b_1}{b_2}$ = $\frac{c_1}{c_2}$ i.e.,

$\frac{2}{2a}$= $\frac{3}{a+b}$= $\frac{-7}{-28}$

⇒$\frac{1}{a}$= $\frac{1}{4}$ and $\frac{3}{a+b}$= $\frac{1}{4}$

⇒ a = 4 and $\frac{3}{4+b}$= $\frac{1}{4}$

⇒ a = 4 and 12=4+b

⇒ a = 4 and b = 8 , are the required values for the given equations of line to have infinitely many solutions.