To find the two numbers such that their product is zero and one number is three more than the other, we can follow these step-by-step instructions:
1. Define the variables:
- Let's call one of the numbers [tex]\( x \)[/tex].
- The other number, being 3 more than [tex]\( x \)[/tex], can be written as [tex]\( y = x + 3 \)[/tex].
2. Set up the equation:
- Since the product of the two numbers is zero, we have:
[tex]\[
x \cdot y = 0
\][/tex]
Substituting [tex]\( y = x + 3 \)[/tex] into the equation, we get:
[tex]\[
x \cdot (x + 3) = 0
\][/tex]
3. Solve the equation:
- Now we need to solve the equation for [tex]\( x \)[/tex]:
[tex]\[
x(x + 3) = 0
\][/tex]
- This equation can be factored, giving us:
[tex]\[
x = 0 \quad \text{or} \quad x + 3 = 0
\][/tex]
4. Find the solutions for [tex]\( x \)[/tex]:
- From [tex]\( x = 0 \)[/tex], we get one solution, [tex]\( x = 0 \)[/tex].
- From [tex]\( x + 3 = 0 \)[/tex], solving for [tex]\( x \)[/tex]:
[tex]\[
x = -3
\][/tex]
5. Determine the corresponding [tex]\( y \)[/tex] values:
- For [tex]\( x = 0 \)[/tex], we substitute it back into [tex]\( y = x + 3 \)[/tex]:
[tex]\[
y = 0 + 3 = 3
\][/tex]
- For [tex]\( x = -3 \)[/tex], we substitute it back into [tex]\( y = x + 3 \)[/tex]:
[tex]\[
y = -3 + 3 = 0
\][/tex]
6. List the pairs of numbers:
- The pairs of numbers that satisfy the given conditions are [tex]\( (0, 3) \)[/tex] and [tex]\( (-3, 0) \)[/tex].
Therefore, the two numbers are:
- [tex]\( (-3, 0) \)[/tex]
- [tex]\( (0, 3) \)[/tex]
These pairs show the two distinct sets of numbers where their product equals zero, and one number is three more than the other.
