Question

if the system of equations 2x+3y=7 and 2ax+(a+b) y=28 has infinitely many solutions

Answer 1

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


if there are any infinite solns, then it means that the lines completely overlap. That is they're the same.

so their coefficients are in proportion.

2/2a = 3/a+b = 1/4

from 1 and 3

a=4

3/4+b = 1/4

b= 8



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Answer 2

Given :

    $2x=3y=7$

   $2ax+(a+b)y=28$

 Comparing $2x+3y=7$ with $a_1x +b_1y+c_1$ =0,

 we get,  $a_1=2, b_1=3, c_1=-7$

comparing $2ax+(a+b)y=28$  with $a_2x+b_2y+c_2 = 0$

$a_2 = 2a, b_2=(a+b), c_2= -28$

For infinitely many solutions, we know,

$\frac{a_1}{a_2} =\frac{b_1}{b_2} =\frac{c_1}{c_2}$

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

 $\frac{2}{2a}=\frac{3}{a+b}$

⇒ $6a = 2a+2b$

⇒ $4a = 2b$

⇒$2a = b$