To determine the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy the conditions [tex]\( |x| = 16 \)[/tex], [tex]\( |y| = 16 \)[/tex], and [tex]\( x + y = 0 \)[/tex], let's go through the requirements step-by-step:
1. Investigate [tex]\( |x| = 16 \)[/tex]:
- The absolute value of [tex]\( x \)[/tex] being 16 means [tex]\( x \)[/tex] can be either 16 or -16.
Therefore, the possible values for [tex]\( x \)[/tex] are:
[tex]\[
x = 16 \quad \text{or} \quad x = -16
\][/tex]
2. Investigate [tex]\( |y| = 16 \)[/tex]:
- Just like with [tex]\( x \)[/tex], if the absolute value of [tex]\( y \)[/tex] is 16, then [tex]\( y \)[/tex] can be either 16 or -16.
Therefore, the possible values for [tex]\( y \)[/tex] are:
[tex]\[
y = 16 \quad \text{or} \quad y = -16
\][/tex]
3. Find pairs [tex]\((x, y)\)[/tex] such that [tex]\( x + y = 0 \)[/tex]:
- We need the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] such that their sum is zero.
4. Check each combination:
- Let’s list all possible combinations of the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
- If [tex]\( x = 16 \)[/tex] and [tex]\( y = 16 \)[/tex]:
[tex]\[
16 + 16 = 32 \quad (\text{not zero})
\][/tex]
- If [tex]\( x = 16 \)[/tex] and [tex]\( y = -16 \)[/tex]:
[tex]\[
16 + (-16) = 0 \quad (\text{satisfies the condition})
\][/tex]
- If [tex]\( x = -16 \)[/tex] and [tex]\( y = 16 \)[/tex]:
[tex]\[
-16 + 16 = 0 \quad (\text{satisfies the condition})
\][/tex]
- If [tex]\( x = -16 \)[/tex] and [tex]\( y = -16 \)[/tex]:
[tex]\[
-16 + (-16) = -32 \quad (\text{not zero})
\][/tex]
5. Determine valid pairs [tex]\((x, y)\)[/tex]:
- From the combinations above, the pairs that satisfy [tex]\( x + y = 0 \)[/tex] are:
[tex]\[
(x, y) = (16, -16) \quad \text{and} \quad (x, y) = (-16, 16)
\][/tex]
Thus, the valid values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy all the conditions given are:
[tex]\[
x = 16, \, y = -16 \quad \text{or} \quad x = -16, \, y = 16
\][/tex]
These pairs effectively result in their sum being zero while also satisfying the absolute value conditions.
