Question

do only equations with variables on both sides have no solution? or can the variable be just on one side? justify your answer. choose the correct answer below. a. any equation can have one solution, infinitely many solutions, or no solutions. this does not depend on whether the variable appears on only one side or on both sides. b. every equation with variables on both sides has no solution. however, it is also possible for an equation with the variable on only one side to also have no solution. c. if the variable only appears on one side of an equation, then the equation can always be solved for the variable. this means that if an equation has no solution, then the variable must appear on both sides. d. if the variable only appears on one side of an equation, then the equation can always be solved for the variable. by contrast, an equation with variables on both sides always has no solution or infinitely many solutions.

Explanation:

To determine the correct answer, we analyze each option:

• Option A: Equations can have 1 solution, infinitely many, or no solution, regardless of variable placement (e.g., \(2x = 4\) has 1 solution, \(x = x\) has infinitely many, \(x = x + 1\) has no solution; variable on both sides in some, one side in others). This matches the description.
• Option B: Equations with variables on both sides can have solutions (e.g., \(2x + 3 = x + 5\) has a solution \(x = 2\)), so this is incorrect.
• Option C: An equation with no solution (e.g., \(x + 1 = x + 2\)) has the variable on both sides, not one side, so this is incorrect.
• Option D: Equations with variables on one side can have no solution? No, e.g., \(x + 1 = 0\) has a solution (\(x=-1\)), and equations with variables on both sides can have 1 solution (e.g., \(2x = x + 1\) has \(x = 1\)), so this is incorrect.

Answer:

A. Any equation can have one solution, infinitely many solutions, or no solutions. This does not depend on whether the variable appears on only one side or on both sides.