Which statement correctly distinguishes a single-variable linear expression from a linear expression with multiple variables?

Which statement correctly distinguishes a single-variable linear expression from a linear expression with multiple variables?

• A. A single-variable linear expression has exactly one variable with no exponent; a linear expression can have multiple variables with no exponents.
• B. A single-variable linear expression has two variables; a linear expression has one variable.
• C. A single-variable linear expression is always a constant; a linear expression is never constant.
• D. A single-variable linear expression must be irrational; a linear expression is rational.

Answer: (A)

The distinction rests on how many variables appear and that each variable is only to the first power. A linear expression in one variable looks like ax + b: there is exactly one variable and it isn’t raised to any higher power. When more variables are allowed, you can have expressions like ax + by + c, which still stay linear because every variable appears to the first power and there are no products of variables. So the statement that the single-variable case has exactly one variable with no exponent, while a linear expression can include multiple variables with no exponents, best captures this difference. Remember, a single-variable linear expression isn’t necessarily constant (it changes with x if the coefficient a isn’t zero), and a multi-variable linear expression can be constant in the degenerate case where all coefficients are zero; the essential point is the number of variables and the first-power form.

The distinction rests on how many variables appear and that each variable is only to the first power. A linear expression in one variable looks like ax + b: there is exactly one variable and it isn’t raised to any higher power. When more variables are allowed, you can have expressions like ax + by + c, which still stay linear because every variable appears to the first power and there are no products of variables. So the statement that the single-variable case has exactly one variable with no exponent, while a linear expression can include multiple variables with no exponents, best captures this difference. Remember, a single-variable linear expression isn’t necessarily constant (it changes with x if the coefficient a isn’t zero), and a multi-variable linear expression can be constant in the degenerate case where all coefficients are zero; the essential point is the number of variables and the first-power form.