Which equation can pair with 3x + 4y = 8 to create a consistent and independent system?

Solution:

The given question is based on the nature of solutions of a system of simultaneous linear equations.

A system of two linear equations:

\((a)_{1}\)x + \((b)_{1}\)y + \((c)_{1}\) = 0 and \((a)_{2}\)x + \((b)_{2}\)y + \((c)_{2}\) = 0 is said to be a consistent and independent system if and only if \((a)_{1}\)/\((a)_{2}\) ≠ \((b)_{1}\)/\((b)_{2}\)

Given 3x + 4y = 8 ⇒ \((a)_{1}\) = 3 , \((b)_{1}\) = 4, \((c)_{1}\) = 8

For a system of linear equations to be a consistent and independent system \((a)_{1}\)/\((a)_{2}\) ≠ \((b)_{1}\)/\((b)_{2}\)

⇒ \((a)_{1}\)/\((b)_{1}\) ≠ \((a)_{2}\)/\((b)_{2}\)

⇒ \((a)_{2}\)/\((b)_{2}\) ≠ 3/4

⇒ \((a)_{2}\)/\((b)_{2}\) can be 5/8 or 3/5 or any fraction which is not an equivalent fraction of 3/4.

Thus, 5x + 8y = 12 or any equation of form Ax + By = C with 4A ≠ 3B can be paired with 3x + 4y = 8 to create a consistent and independent system.

Which equation can pair with 3x + 4y = 8 to create a consistent and independent system?

Summary:

Ax + By = C with 4A ≠ 3B can be paired with 3x + 4y = 8 to create a consistent and independent system. Example: 5x + 8y = 12.