Answer:
[tex]\text{Option B.}[/tex]
Step-by-step explanation:
[tex]\text{If you simplify each equations in the given options into the form of the}\\\text{equation given in the question i.e. in the form }Ax+By=C\text{ and get the}\\\text{same equation, then you will have infinite number of solutions.}[/tex]
[tex]\text{We have,}\\3x-2y=10\\\text{Slope = }-\dfrac{\text{coefficient of }x}{\text{coefficient of }y}=\dfrac{3}{2}[/tex]
[tex]\bold{Option\ A:}\\y=-\dfrac{3}{2}x+1\\\\\text{or, }y-1=-\dfrac{3}{2}x\\\\\text{or, }2y-2=-3x\\\\\text{or, }3x+2y=2[/tex]
[tex]\bold{Option\ B:}\\y=\dfrac{3}{2}x-5\\\\\text{or, }y+5=\dfrac{3}{2}x\\\\\text{or, }2y+10=3x\\\text{or, }3x-2y=10\text{, which is exactly same as the one given in the equation.}\\\text{So this equation would have infinite solutions in a system of equations with:}\\3x-2y=10.[/tex]
[tex]\bold{Option\ C:}\\y=\dfrac{2}{3}x+5\\\\\text{or, }y-5=\dfrac{2}{3}x\\\\\text{or, }3y-15=2x\\\text{or, }2x-3y=-15\\\text{This is not the same equation as }3x-2y=10,\text{ so the system of equations}\\\text{would give finite number of solutions.}[/tex]
[tex]\bold{Option\ D}\\\\y=-\dfrac{2}{3}x-1\\\\\text{or, }y+1=-\dfrac{2}{3}x\\\text{or, }3y+3=-2x\\\text{or, }2x+3y=-3\\\text{Also, this is a different equation. So it will give only finite number of}\\\text{solutions with }3x-2y=10.[/tex]
[tex]\text{Parallel lines have no solution; coincident lines have infinite number of}\\\text{solutions. Because parallel lines never intersect, but coincident lines line up}\\\text{exactly with each other so that all points on the two lines are the same.}[/tex]
[tex]\text{For example, }3x-2y=10\text{ and }3x-2y=5\text{ will have no solutions because }\\\text{they are parallel lines.}[/tex]
