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 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 fraction or a negative number.
d. the statement is never true because the solution involves division, meaning ( x ) will be a fraction.

Explanation:

First, solve the equation \( ax + b = c \) for \( x \).

Step 1: Subtract \( b \) from both sides

To isolate the term with \( x \), we subtract \( b \) from both sides of the equation. This gives us \( ax = c - b \).

Step 2: Divide both sides by \( a \) (assuming \( a

eq 0 \))
To solve for \( x \), we divide both sides of the equation \( ax = c - b \) by \( a \). So, \( x=\frac{c - b}{a} \).

Now, analyze the statement: "If \( a \), \( b \), and \( c \) are whole numbers, \( x \) is a whole number."

• Let's take an example where \( a = 2 \), \( b = 1 \), \( c = 3 \). Then \( x=\frac{3 - 1}{2}=\frac{2}{2} = 1 \), which is a whole number.
• Now take another example where \( a = 2 \), \( b = 1 \), \( c = 2 \). Then \( x=\frac{2 - 1}{2}=\frac{1}{2} \), which is not a whole number. Also, if \( a = 1 \), \( b = 3 \), \( c = 2 \), then \( x=\frac{2 - 3}{1}=- 1 \), which is not a whole number (whole numbers are non - negative integers).

So the solution \( x=\frac{c - b}{a} \) can be a fraction (when \( c - b \) is not divisible by \( a \)) or a negative number (when \( c - b\lt0 \) and \( a\gt0 \)). So the statement "If \( a \), \( b \), and \( c \) are whole numbers, \( x \) is a whole number" is sometimes true because the solution could also be a fraction or a negative number.

Answer:

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