Question

If ad≠bc, then find whether the pair of linear equations ax+by=p and cx +dy =p has no solutions, unique solutions or infinitely many solutions.

Answer 1

Hope this is helpful!

Answer 2

The pair of linear equations ax+by=p and cx +dy =p has unique solutions.

Step-by-step explanation:

1. Given linear equation are

  $\mathbf{ax+by-p=0}$        ...1)

  $\mathbf{cx+dy-p=0}$        ...2)

2. From cross multiplication method

                x                          y                       1

  b                         -p                        a                       b

  d                         -p                        c                       d

   We can write

  $\mathbf{\frac{x}{(b)\times (-p)-(d)\times (-p)}=\frac{y}{(-p)\times (c)-(-p)\times (a)}=\frac{1}{(a)\times (d)-(c)\times (b)}}$      ...3)

3. After solving denominator of equation 3), we get

 $\mathbf{\frac{x}{-bp+pd}=\frac{y}{-pc+ap}=\frac{1}{ad-bc}}$

 So value of x and y are

  $\mathbf{x=\frac{-bp+pd}{ad-bc}}$     and $\mathbf{y=\frac{-pc+ap}{ad-bc}}$

 but here is given that   ad ≠ bc so value of x and y are exist. It means its have unique solutions.