Answer question 9 :-
Answers
Answer:
a = 5 and b = 1 is the value of the pair of linear equations \bold{2x + 3y = 7}2x+3y=7 and \bold{a(x + y) -b(x-y) = 3a +b -2}a(x+y)−b(x−y)=3a+b−2
Given:
2x + 3y = 72x+3y=7
a(x+y) -b(x-y) = 3a +b -2a(x+y)−b(x−y)=3a+b−2
To find:
Value of a and b =?
Solution:
For the equations to have infinite number of solutions, the ratios of the coefficients of x and y and constants in two equations should be equal.
That is, \frac{2}{a-b}=\frac{3}{a+b}=\frac{7}{3 a+b-2}
a−b
2
=
a+b
3
=
3a+b−2
7
Let us take two equations at a time to solve for a and b\frac{2}{a-b}=\frac{3}{a+b}
a−b
2
=
a+b
3
\begin{lgathered}\begin{array}{l}{\Rightarrow 2(a+b)=3(a-b)} \\ {\Rightarrow 2 a+2 b=3 a-3 b} \\ {\Rightarrow a=5 b}\end{array}\end{lgathered}
⇒2(a+b)=3(a−b)
⇒2a+2b=3a−3b
⇒a=5b
Now,
\begin{lgathered}\begin{array}{l}{\frac{2}{a-b}=\frac{7}{3 a+b-2}} \\ {2(3 a+b-2)=7(a-b)}\end{array}\Rightarrow 6 a+2 b-4=7 a-7 b\end{lgathered}
a−b
2
=
3a+b−2
7
2(3a+b−2)=7(a−b)
⇒6a+2b−4=7a−7b
a-9b+4=0a−9b+4=0
Substitute a=5ba=5b in the above equation
\begin{lgathered}\begin{array}{l}{5 b-9 b+4=0} \\ {-4 b+4=0} \\ {b=\frac{4}{4}=1} \\ {a=5 b=5(1)=5}\end{array}\end{lgathered}
5b−9b+4=0
−4b+4=0
b=
4
4
=1
a=5b=5(1)=5
Therefore, \bold{a=5 and b=1}a=5andb=1 is the value of a and b.