To determine which equation [tex]\( x \)[/tex] and [tex]\( y \)[/tex] satisfy, we will analyze the given options and ultimately reach our conclusion.
Let's revisit the equations given in the options:
a. [tex]\(3x + 8y = 0\)[/tex]
b. [tex]\(3x - 8y = 0\)[/tex]
c. [tex]\(8x + 3y = 0\)[/tex]
d. [tex]\(8x = 3y\)[/tex]
Now, let's consider the logical consistency of each option:
1. Option a: [tex]\(3x + 8y = 0\)[/tex]
This equation indicates that the linear combination of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] with coefficients 3 and 8 must equal zero. This means that for any pair [tex]\((x, y)\)[/tex] satisfying this relation, shifting one variable would require a corresponding shift in the other variable to maintain the relationship.
2. Option b: [tex]\(3x - 8y = 0\)[/tex]
This equation suggests that [tex]\(3x\)[/tex] is equal to [tex]\(8y\)[/tex], with opposing signs. Similarly, this would mean there is a linear relationship but one being positive and the other being negative.
3. Option c: [tex]\(8x + 3y = 0\)[/tex]
Like option a, this option requires a balance between [tex]\( x \)[/tex] and [tex]\( y \)[/tex] but with different coefficients.
4. Option d: [tex]\(8x = 3y\)[/tex]
This equation is simply another form representing a direct proportionality between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
After careful consideration, option (a) [tex]\(3x + 8y = 0\)[/tex] stands out as the most logically consistent equation for the given problem, meaning [tex]\( x \)[/tex] and [tex]\( y \)[/tex] satisfy this relationship perfectly.
Hence, the correct answer is:
Option (a) [tex]\( 3x + 8y = 0 \)[/tex]
