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. 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. a. the statement is sometimes true because the solution could also be a fraction. b. the statement is sometimes true because the solution could also be a fraction but not a negative number. c. the statement is sometimes true because the solution could also be a negative number. d. the statement is never true because the solution involves division, meaning x will be a fraction. b. if a, b, and c are integers, x is an integer. choose the correct answer below a. the statement is sometimes true because the solution could also be a fraction. 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 negative number. d. the statement is always true. the solution for x is an expression involving addition, subtraction, multiplication, and/or division, and these operations always give integers when a, b, and c are integers.

Explanation:

Part 1: Solve for \( x \) in \( ax + b = c \)

Step 1: Isolate the term with \( x \)

Subtract \( b \) from both sides of the equation \( ax + b = c \) to get \( ax = c - b \).

Step 2: Solve for \( x \)

Divide both sides by \( a \) (assuming \( a
eq 0 \)): \( x=\frac{c - b}{a} \).

The solution \( x=\frac{c - b}{a} \). If \( a = 2 \), \( b = 1 \), \( c = 3 \), then \( x=\frac{3 - 1}{2}=1 \) (whole number). But if \( a = 2 \), \( b = 1 \), \( c = 4 \), then \( x=\frac{4 - 1}{2}=\frac{3}{2} \) (not a whole number). So the statement is sometimes true. Option C says "The statement is sometimes true because the solution could also be a fraction or a negative number" – but our example shows fraction, and negative is also possible (e.g., \( a = 1 \), \( b = 3 \), \( c = 1 \), \( x=1 - 3=-2 \)). So the correct option is C.

Given \( a, b, c \) are integers, \( x=\frac{c - b}{a} \). If \( a = 2 \), \( b = 1 \), \( c = 3 \), \( x = 1 \) (integer). If \( a = 2 \), \( b = 1 \), \( c = 4 \), \( x=\frac{3}{2} \) (not integer). So the statement "The statement is sometimes true because the solution could also be a fraction" (Option A) is correct. Option B says "never true" (wrong, as we have integer solutions). Option C mentions negative (but the question is about \( x \) being integer, not sign). Option D says "always true" (wrong, as fraction solutions exist).

Answer:

\( x = \frac{c - b}{a} \) (for \( a
eq0 \))

Part a: Analyze the statement