Question

2.
The value of a for which the pair of equations 10x + 5y = a -5, 20x + 10y - a=0 has infinitely many
(a) 5
loy-a=0 has infinitely many solutions is
(b)-10
(c) 10
(d) 20​

Answer 1

The value of a is (c) 10

Step-by-step explanation:

Given:

the equations are 10x+5y = a-5, 20x + 10y -a=0

To find:

value of a for which the equations have infinitely many solutions

Solution:

The first equation will be

10x + 5y - (a-5) = 0

The other equation is 20x+10y-a=0

Now, comparing the equations to the standard form

$a_{1}x +b_{1}y +c_{1} \ \ \\ \ \\ a_{2}x +b_{2}y +c_{2}$

we get, $a_{1}= 10, b_{1}= 5, c_{1} = -(a-5)$

$a_{2}= 20,b_{2}= 10 ,c_{2}= -a$

The pair of equations has infinitely many solutions

So, $\frac{a_{1} }{a_{2}} =\frac{b_{1}}{b_{2}} =\frac{c_{1}}{c_{2}}$

∴ $\frac{10}{20} = \frac{5}{10} =\frac{-(a-5)}{-a}$

∴ $\frac{10}{20} = \frac{5}{10} =\frac{a-5}{a}$

Using the equality

∴ $\frac{5}{10} =\frac{a-5}{a}$

∴ $5a = 10 ( a -5 )$

∴ 5a = 10 (a-5)

∴ 5a = 10a - 50

∴ 10a - 50 = 5a

∴ 10a-5a = 50

∴ 5a = 50

∴ a =50/5

∴ a = 10

Thus, the value of a for the pair equations  10x+5y= a -5, 20x+10y-a=0 having infinitely many solutions is 10

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