Question

on a number line, a number, b, is located the same distance from 0 as another number, a, but in the opposite direction. the number b varies directly with the number a. for example ( b = 2\frac{3}{4} ) when ( a = -2\frac{3}{4} ). which equation represents this direct variation between a and b?
options:
( b = -a )
( b(-a) = 0 )
( b - a = 0 )
( -b = -a )

Explanation:

Step1: Analyze the relationship between a and b

Given that \( b \) is the same distance from 0 as \( a \) but in the opposite direction. So if \( a \) is a number, \( b \) should be the negative of \( a \), i.e., \( b=-a \). Let's check with the example: when \( a = - 2\frac{3}{4}\), then \( b=-a=-(-2\frac{3}{4}) = 2\frac{3}{4}\), which matches the given example.

Step2: Check other options

• For \( b(-a)=0 \): If \( b=-a \), then \( b(-a)=(-a)(-a)=a^{2}

eq0 \) (unless \( a = 0 \)), so this is incorrect.

• For \( b - a=0 \): This implies \( b=a \), but \( b \) is in the opposite direction, so this is incorrect.
• For \( -b=-a \): This implies \( b=a \), same as above, incorrect.

Answer:

\( b = -a \)