Question

determine whether each statement is true or false.
a linear equation has one solution when the variable equals zero.
a linear equation has infinitely many solutions when both sides of the equation are equivalent.
a linear equation has one solution when the equation makes a false statement.

Explanation:

To solve this, we analyze each statement about linear equations:

Statement 1: "A linear equation has ONE solution when the variable equals zero."

A linear equation in one variable is of the form \(ax + b = 0\) (\(a
eq0\)). The solution is \(x = -\frac{b}{a}\). The solution being zero is just one specific case (when \(b = 0\)), but a linear equation (with \(a
eq0\)) always has exactly one solution, regardless of what that solution is. However, the statement implies that the act of the variable equaling zero is what gives it one solution, which is a misrepresentation. A linear equation (non - identity, non - contradiction) always has one solution, not because the variable equals zero. So this statement is False.

Statement 2: "A linear equation has INFINITELY MANY solutions when both sides of the equation are equivalent."

If both sides of a linear equation are equivalent (for example, \(2x+4 = 2(x + 2)\)), when we simplify, we get a statement like \(0=0\) (or \(a=a\) for some real number \(a\)). This means that any value of the variable will satisfy the equation, so there are infinitely many solutions. So this statement is True.

Statement 3: "A linear equation has ONE solution when the equation makes a false statement."

If a linear equation makes a false statement (for example, \(2x+3=2x + 5\), which simplifies to \(3 = 5\)), this is a contradiction. A contradiction has no solutions, not one solution. So this statement is False.

Answer:

s:

• "A linear equation has ONE solution when the variable equals zero." : False
• "A linear equation has INFINITELY MANY solutions when both sides of the equation are equivalent." : True
• "A linear equation has ONE solution when the equation makes a false statement." : False