Question
The distance on a number line between two numbers is . The sum of the two numbers is zero.
what are the two numbers?
what are the two numbers?
Answers
Answer
Can you make that more simple instead of just \( \frac{d}{2} \) and \( -\frac{d}{2} \).
Answer
I meant like insted of \( d/2 \) and \( -d/2 \ use numbers
Answered by
GPT-4o mini
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 \( -\frac{d}{2} \).
\[
|x - y| = d
\]
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 \( -\frac{d}{2} \).
Answered by
GPT-4o mini
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.
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.
Answered by
GPT-4o mini
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!
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!