Question

The distance on a number line between two numbers is . The sum of the two numbers is zero.
what are the two numbers?

Answer 1

Can you make that more simple instead of just \( \frac{d}{2} \) and \( -\frac{d}{2} \).

Answer 2

I meant like insted of \( d/2 \) and \( -d/2 \ use numbers

Answer 3

Let the two numbers be \( x \) and \( y \). According to the problem, the distance between the two numbers on a number line is represented mathematically as:

\[
|x - y| = d
\]

where \( d \) is the distance between the two numbers. Additionally, we are given that the sum of the two numbers is zero:

\[
x + y = 0
\]

From the equation \( x + y = 0 \), we can express one number in terms of the other. Let's express \( y \) in terms of \( x \):

\[
y = -x
\]

Now, substituting \( y = -x \) into the distance equation, we have:

\[
|x - (-x)| = d
\]
\[
|x + x| = d
\]
\[
|2x| = d
\]

This simplifies to:

\[
2|x| = d
\]

From here, we can solve for \( |x| \):

\[
|x| = \frac{d}{2}
\]

This gives us two possibilities for \( x \):

1. \( x = \frac{d}{2} \)
2. \( x = -\frac{d}{2} \)

Now, we can find \( y \) using \( y = -x \):

1. If \( x = \frac{d}{2} \), then \( y = -\frac{d}{2} \).
2. If \( x = -\frac{d}{2} \), then \( y = \frac{d}{2} \).

In both cases, the two numbers are:

\[
\left(\frac{d}{2}, -\frac{d}{2}\right) \quad \text{or} \quad \left(-\frac{d}{2}, \frac{d}{2}\right)
\]

Thus, the two numbers are \( \frac{d}{2} \) and \( -\frac{d}{2} \).

Answer 4

Certainly! Let's simplify the expression for the two numbers without using fractions.

Given:
- The sum of the two numbers \( x \) and \( y \) is zero:
\( x + y = 0 \).

From this, we can deduce that:
- \( y = -x \).

Also, we know that the distance between the two numbers is \( d \):
- The distance can be expressed as:
\( |x - y| = d \).

Substituting \( y = -x \) into the distance equation gives us:
\[
|x - (-x)| = d \implies |x + x| = d \implies |2x| = d.
\]

This implies:
\[
2 |x| = d \implies |x| = \frac{d}{2}.
\]

From this, we can determine the two possible values for \( x \):
1. \( x = \frac{d}{2} \)
2. \( x = -\frac{d}{2} \)

Now, substituting back to find \( y \):
- If \( x = \frac{d}{2} \), then \( y = -\frac{d}{2} \).
- If \( x = -\frac{d}{2} \), then \( y = \frac{d}{2} \).

To state it simply, the two numbers can be expressed as:
- **If the distance \( d \) is positive**: the two numbers are \( \frac{d}{2} \) and \( -\frac{d}{2} \).
- In common language, you can think of them as a certain distance from zero, one positive and one negative.

Thus, the specific values of the two numbers are:
- **The two numbers are \( d/2 \) and \( -d/2 \)**, where \( d \) is the specified distance.

If you want to express them in a way that emphasizes their properties:
- They can be seen as being equidistant from zero in opposite directions.

Answer 5

Got it! Let's use specific numbers to illustrate the concept.

Let's say the distance \( d \) is 6.

Using the relationships we derived earlier:

1. The two numbers are \( \frac{d}{2} \) and \( -\frac{d}{2} \).

Substituting \( d = 6 \):

- \( \frac{6}{2} = 3 \)
- \( -\frac{6}{2} = -3 \)

So, in this case, the two numbers are:
- \( 3 \) and \( -3 \).

**To generalize**: For any positive distance \( d \), the two numbers will always be \( \frac{d}{2} \) and \( -\frac{d}{2} \).

For a different example, if \( d = 10 \):
- The two numbers would be \( 5 \) and \( -5 \).

If you have a specific distance in mind, just divide that by 2, and you'll have your two numbers!