Final answer:
In the context of the linear equations provided, equation d) represents infinitely many solutions as it is reducible to a simpler form that is equivalent to the original equation. The remaining equations, a), b), and c), are distinct from each other and thus would have one unique solution when each is paired with any other non-parallel line.
Step-by-step explanation:
When identifying which equations have one solution, infinitely many solutions, or no solution, we are examining the linear equations and their properties. The provided equations are:
- a) 3x - 2y = 5
- b) 2x + 4y = 10
- c) 2x - 2y = 6
- d) 4x - 8y = 16
To determine the number of solutions for each equation:
- Put each equation in slope-intercept form (y = mx + b), if possible, to identify unique slopes and y-intercepts.
- If two lines have the same slope but different y-intercepts, they are parallel and do not intersect, indicating no solution.
- If two lines have the same slope and the same y-intercept, they are coincident, indicating infinitely many solutions.
- If two lines have different slopes, they intersect at exactly one point, indicating one solution.
For the equation d) 4x - 8y = 16, if we divide through by 4, we receive x - 2y = 4, which is identical to the original standard form after multiplication by a nonzero constant. Therefore, this equation represents infinitely many solutions. The other equations, a), b) and c), are not multiples of each other; hence each represents a different line that will have one unique solution where they intersect with any other non-parallel line.