Given a linear equation in two variables: 5x + 7y - 8 = 0, which of the following linear equations, along with the given equation, forms a system of linear equations having no solution?

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Accepted Answer

To determine which equation forms a system with no solution along with the given equation 5x + 7y - 8 = 0, we need to check for parallel lines. Two linear equations have no solution if their lines are parallel, i.e., their coefficients of x and y are proportional, but the constants are not.

Given equation: 5x + 7y - 8 = 0

Check each option:

1) 5x + 7y - 16 = 0
   Coefficients ratio: 5/5 = 1, 7/7 = 1
   Constants ratio: -8/-16 = 0.5
   Since coefficients ratio are equal but constants ratio is different, lines are parallel => no solution.

2) 7x + 5y - 8 = 0
   Coefficients ratio: 5/7 ≠ 7/5, not proportional => lines intersect => solution exists.

3) 5x - 7y - 8 = 0
   Coefficients ratio: 5/5 = 1, 7/-7 = -1, not equal => lines intersect => solution exists.

4) 10x + 14y - 16 = 0
   Coefficients ratio: 5/10 = 0.5, 7/14 = 0.5
   Constants ratio: -8/-16 = 0.5
   All ratios equal, so equations are dependent (same line), infinite solutions, not no solution.

Therefore, option 1 forms a system with no solution.

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