Write the set of values of a and b for which the following system of equations has infinitely many solutions.
Write the set of values of a and b for which the following system of equations has infinitely many solutions.
$2 x+3 y=7$
$2 a x+(a+b) y=28$
The given equations are
$2 x+3 y-7=0$
$2 a x+(a+b) y-28=0$
For the equations to have infinite number of solutions,
$\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}$
Therefore
$\frac{2}{2 a}=\frac{3}{a+b}=\frac{7}{28}$
Let us take
$\frac{2}{2 a}=\frac{3}{a+b}$
$2(a+b)=2 a \times 3$
$2 a+2 b=6 a$
$0=6 a-2 a-2 b$
$0=4 a-2 b$
$\frac{3}{a+b}=\frac{7}{28}$
$28 \times 3=7(a+b)$
$84=7 a+7 b$
By dividing both the sides by 7 we get,
$12=a+b \cdots(i i)$
By multiplying equations $($ ii $)$ by 2 we get
$24=2 a+2 b \cdots(i i i)$
Substituting $(i i i)$ from $(i)$ we get
$a=\frac{24}{6}$
$a=4$
Subtracting $a=4$ in equation (iii) we have
$24=2 a+2 b$
$24=2 \times 4+2 b$
$24=8+2 b$
$24-8=2 b$
$16=2 b$
$\frac{16}{2}=b$
$8=b$
Hence, the value of $a=4, b=8$ when system of equations has infinity many solutions.
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