For which values of $a$ and $b$, will the following pair of linear equations have infinitely many solutions?$x+2 y=1$$(a-b) x+(a+b) y=a+b-2$

Complete Python Prime Pack for 2023

9 Courses     2 eBooks

Artificial Intelligence & Machine Learning Prime Pack

6 Courses     1 eBooks

Java Prime Pack 2023

9 Courses     2 eBooks

Given:

Equations: $x+2y=1$; $(a−b)x+(a+b)y=2$

To do:

We have to find the values of $a$ and $b$, for which the following pair of linear equations will have infinitely many solutions.

Solution:

The condition for infinitely many solutions is,

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

​$\Rightarrow\frac{1}{( a-b)}=\frac{2}{( a+b)}=\frac{1}{a+b-2}$

From ratio I and II

$2a-2b=a+b$

$\Rightarrow a-3b=0\ ....( i)$

From ratio II and III

$2a+2b-4=a+b$

$\Rightarrow a+b=4\ .....( ii)$

Now solving $( i)$ and $( ii)$ we have

$a-3b=0\ ......( i)$

$a+b=4\ ......( ii)$  [Subtracting $( ii)$ from $( i)$ ]

$-4b=-4$

$\Rightarrow b=1$

and $a=4-b$

$\Rightarrow a=4-1$         [from $( ii)$]

$\Rightarrow a=3$

The values of $a$ and $b$ are $3$ and $1$ respectively.

Updated on 10-Oct-2022 13:27:14