Let's solve the problem step-by-step:
1. Define the variables:
Let's call the two numbers [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
2. Translate the problem into equations:
- The difference of the two numbers is [tex]\(-5\)[/tex]:
[tex]\[
x - y = -5 \quad \text{(Equation 1)}
\][/tex]
- The sum of their squares is [tex]\(17\)[/tex]:
[tex]\[
x^2 + y^2 = 17 \quad \text{(Equation 2)}
\][/tex]
3. Solve Equation 1 for one of the variables:
Let's solve Equation 1 for [tex]\( x \)[/tex]:
[tex]\[
x - y = -5 \implies x = y - 5 \quad \text{(Equation 3)}
\][/tex]
4. Substitute Equation 3 into Equation 2:
Substitute [tex]\( x = y - 5 \)[/tex] into [tex]\( x^2 + y^2 = 17 \)[/tex]:
[tex]\[
(y - 5)^2 + y^2 = 17
\][/tex]
Expand and simplify:
[tex]\[
(y^2 - 10y + 25) + y^2 = 17
\][/tex]
Combine like terms:
[tex]\[
2y^2 - 10y + 25 = 17
\][/tex]
Subtract 17 from both sides:
[tex]\[
2y^2 - 10y + 8 = 0
\][/tex]
Simplify the quadratic equation (divide all terms by 2):
[tex]\[
y^2 - 5y + 4 = 0
\][/tex]
5. Solve the quadratic equation:
Factor the quadratic equation:
[tex]\[
(y - 4)(y - 1) = 0
\][/tex]
So, the solutions for [tex]\( y \)[/tex] are:
[tex]\[
y = 4 \quad \text{or} \quad y = 1
\][/tex]
6. Find the corresponding [tex]\( x \)[/tex] values:
Use Equation 3, [tex]\( x = y - 5 \)[/tex], to find [tex]\( x \)[/tex] for each [tex]\( y \)[/tex]:
- If [tex]\( y = 4 \)[/tex]:
[tex]\[
x = 4 - 5 = -1
\][/tex]
- If [tex]\( y = 1 \)[/tex]:
[tex]\[
x = 1 - 5 = -4
\][/tex]
7. List the pairs:
The pairs of numbers that fit the given criteria are:
[tex]\[
(-1, 4) \quad \text{and} \quad (-4, 1)
\][/tex]
Therefore, the pairs of numbers whose difference is [tex]\(-5\)[/tex] and the sum of their squares is [tex]\(17\)[/tex] are [tex]\((-1, 4)\)[/tex] and [tex]\((-4, 1)\)[/tex].
