# Which Equation Can Pair With 3x + 4y = 8 to Create a Consistent and Independent System?

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

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

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

Let us explore more about the consistent and independent system or unique solution case of the system of linear equations.

## Explanation:

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.

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