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

what are the two numbers?

5 answers

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 \( -\frac{d}{2} \).

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

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

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!