To solve the equation [tex]\( |-9y| = 27 \)[/tex], let's go through the steps involved.
1. Understanding Absolute Value:
The absolute value of a number, defined as [tex]\( |x| \)[/tex], is the distance of [tex]\( x \)[/tex] from zero on the number line regardless of direction. For any real number [tex]\( x \)[/tex], [tex]\( |x| = x \)[/tex] if [tex]\( x \geq 0 \)[/tex] and [tex]\( |x| = -x \)[/tex] if [tex]\( x < 0 \)[/tex].
2. Set Up the Equation:
The equation given is [tex]\( |-9y| = 27 \)[/tex]. This means that the expression inside the absolute value, [tex]\( -9y \)[/tex], should be evaluated for both its positive and negative possibilities:
[tex]\[ -9y = 27 \][/tex]
and
[tex]\[ -9y = -27 \][/tex]
3. Solve Each Case Separately:
- Case 1: [tex]\( -9y = 27 \)[/tex]
[tex]\[
-9y = 27
\][/tex]
To solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{27}{-9}
\][/tex]
[tex]\[
y = -3
\][/tex]
- Case 2: [tex]\( -9y = -27 \)[/tex]
[tex]\[
-9y = -27
\][/tex]
To solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{-27}{-9}
\][/tex]
[tex]\[
y = 3
\][/tex]
4. List Both Solutions:
The solutions to the equation [tex]\( |-9y| = 27 \)[/tex] are [tex]\( y = -3 \)[/tex] and [tex]\( y = 3 \)[/tex].
Therefore, the correct choice is [tex]\( y = -3 \)[/tex] and [tex]\( y = 3 \)[/tex].
So the correct answer is:
[tex]\[ y = 3 \text{ and } -3 \][/tex]
