To solve the simultaneous equations:
[tex]\[
\begin{aligned}
x^2 + y^2 &= 13 \\
x &= y - 5
\end{aligned}
\][/tex]
We can follow these steps:
1. Substitute the second equation into the first equation:
Given [tex]\(x = y - 5\)[/tex], we substitute [tex]\(x\)[/tex] in the first equation:
[tex]\[
(y - 5)^2 + y^2 = 13
\][/tex]
2. Expand and simplify:
Expand [tex]\((y - 5)^2\)[/tex]:
[tex]\[
y^2 - 10y + 25 + y^2 = 13
\][/tex]
Combine like terms:
[tex]\[
2y^2 - 10y + 25 = 13
\][/tex]
Subtract 13 from both sides:
[tex]\[
2y^2 - 10y + 12 = 0
\][/tex]
3. Simplify the quadratic equation:
Divide the entire equation by 2 to simplify:
[tex]\[
y^2 - 5y + 6 = 0
\][/tex]
4. Factor the quadratic equation:
We need to find factors of 6 that add up to -5. The factors are -2 and -3.
[tex]\[
(y - 2)(y - 3) = 0
\][/tex]
5. Solve for [tex]\(y\)[/tex]:
Set each factor equal to zero and solve for [tex]\(y\)[/tex]:
[tex]\[
y - 2 = 0 \quad \Rightarrow \quad y = 2
\][/tex]
[tex]\[
y - 3 = 0 \quad \Rightarrow \quad y = 3
\][/tex]
6. Find the corresponding [tex]\(x\)[/tex] values:
Using the second original equation [tex]\(x = y - 5\)[/tex]:
- For [tex]\(y = 2\)[/tex], [tex]\(x = 2 - 5 = -3\)[/tex].
- For [tex]\(y = 3\)[/tex], [tex]\(x = 3 - 5 = -2\)[/tex].
Therefore, the solutions to the simultaneous equations are:
[tex]\[
(x, y) = (-3, 2) \quad \text{and} \quad (x, y) = (-2, 3)
\][/tex]
