Question

px+qy=p-q and qx- py= p+q by substition method



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Answer 1

Answer:

• The value of x and y are 1 and -1 respectively.

Given:

The given equations are

=> px + qy = p - q...(1) and qx - py = p + q...(2)

To find:

• The value of x and y.

Solution:

=> px + qy = p - q....(1)

=> qx - py = p + q....(2)

Squaring equation (1) and equation (2), we get

=> p²x² + q²y² + 2pqxy = p² + q² - 2pq...(3)

=> q²x² + p²y² - 2pqxy = p² + q² + 2pq...(4)

Add equations (1) and (3), we get

=> p²x² + q²x² + q²y² + p²y² = 2 (p² + q²)

=> x² (p² + q²) + y² (p² + q²) = 2 (p² + q²)

Divide throughout by (p² + q²), we get

=> x² + y² = 2...(5)

Multiply equation (1) by p and equation (2) by q, we get

=> p²x + pqy = p (p - q)...(6)

=> q²x - pqy = q (p + q)...(7)

Add equations (6) and (7), we get

=> p²x + q²x = p (p - q) + q (p + q)

=> x (p² + q²) = p² - pq + pq + q²

=> x (p² + q²) = p² + q²

=> x = 1

Substitute x = 1, in equation (5), we get

=> (1)² + y² = 2

=> 1 + y² = 2

=> y² = 2 - 1

=> y² = 1

=> y = 1 and -1

When x = 1 and y = 1, substitute in equation (1), we get

=> L.H.S. = p (1) + q (1) = p + q

But, R.H.S. = p - q, hence the value of y is not equal to 1.

When x = 1 and y = -1, substitute in equation (1), we get

=> L.H.S. = p (1) + q (-1) = p - q = R.H.S.

Hence, the value of y is -1.

The values of x and y are 1 and -1 respectively.