Question

given the equation ax + b = c, solve for x. describe each statement in parts a to c below as always, sometimes, or never true. explain your answer. a. if a, b, and c are whole numbers, x is a whole number. choose the correct answer below. a. the statement is always true. the solution for x is an expression involving addition, subtraction, multiplication, and/or division, and these operations always give whole numbers when a, b, and c are whole numbers. b. the statement is never true because the solution involves division, meaning, x will be a fraction. c. the statement is sometimes true because the solution could also be a fraction but not a negative number. d. the statement is sometimes true because the solution could also be a fraction or a negative number.

Explanation:

Step1: Solve the equation for \( x \)

Given the equation \( ax + b = c \), we first subtract \( b \) from both sides:
\( ax = c - b \)
Then divide both sides by \( a \) (assuming \( a
eq 0 \)):
\( x = \frac{c - b}{a} \)

Step2: Analyze the solution

\( a, b, c \) are whole numbers. The solution \( x = \frac{c - b}{a} \) involves division. For example:

• If \( a = 2 \), \( b = 1 \), \( c = 3 \), then \( x = \frac{3 - 1}{2} = 1 \) (whole number).
• If \( a = 2 \), \( b = 1 \), \( c = 4 \), then \( x = \frac{4 - 1}{2} = \frac{3}{2} \) (fraction, not a whole number).
• If \( a = 3 \), \( b = 5 \), \( c = 2 \), then \( x = \frac{2 - 5}{3} = -1 \) (negative number, not a whole number).

Thus, \( x \) is sometimes a whole number (when \( c - b \) is divisible by \( a \)) and sometimes not (fraction or negative).

Answer:

D. The statement is sometimes true because the solution could also be a fraction or a negative number.