Sure! Let's solve the given system of equations step-by-step. Our goal is to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations:
[tex]\[
\begin{cases}
x^2 - y^2 = 13 \\
x + y = 13
\end{cases}
\][/tex]
Here is the detailed solution:
### Step 1: Solve for one variable in terms of the other from the second equation
From the equation [tex]\( x + y = 13 \)[/tex], we can express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = 13 - x
\][/tex]
### Step 2: Substitute [tex]\( y \)[/tex] into the first equation
Replace [tex]\( y \)[/tex] in the equation [tex]\( x^2 - y^2 = 13 \)[/tex] with [tex]\( 13 - x \)[/tex]:
[tex]\[
x^2 - (13 - x)^2 = 13
\][/tex]
### Step 3: Simplify the equation
Evaluate [tex]\( (13 - x)^2 \)[/tex]:
[tex]\[
(13 - x)^2 = 169 - 26x + x^2
\][/tex]
Now, our equation looks like this:
[tex]\[
x^2 - (169 - 26x + x^2) = 13
\][/tex]
Distribute and simplify:
[tex]\[
x^2 - 169 + 26x - x^2 = 13
\][/tex]
Combine like terms (note that [tex]\( x^2 \)[/tex] cancels out):
[tex]\[
26x - 169 = 13
\][/tex]
### Step 4: Solve for [tex]\( x \)[/tex]
Isolate [tex]\( x \)[/tex]:
[tex]\[
26x = 13 + 169
\][/tex]
[tex]\[
26x = 182
\][/tex]
[tex]\[
x = \frac{182}{26}
\][/tex]
[tex]\[
x = 7
\][/tex]
### Step 5: Find [tex]\( y \)[/tex]
Use the value of [tex]\( x \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[
y = 13 - x
\][/tex]
[tex]\[
y = 13 - 7
\][/tex]
[tex]\[
y = 6
\][/tex]
### Step 6: Verify the solution
Let's verify that [tex]\( x = 7 \)[/tex] and [tex]\( y = 6 \)[/tex] satisfy both original equations:
1. Check [tex]\( x^2 - y^2 = 13 \)[/tex]:
[tex]\[
7^2 - 6^2 = 49 - 36 = 13
\][/tex]
2. Check [tex]\( x + y = 13 \)[/tex]:
[tex]\[
7 + 6 = 13
\][/tex]
Both equations are satisfied. Therefore, the solution to the system of equations is:
[tex]\[
(x, y) = (7, 6)
\][/tex]
Next, we need to consider whether there's another pair that satisfies the equations if we distribute the terms differently. The only other possibility is if we set [tex]\( x + y = -13 \)[/tex] while solving the quadratic form arising from the given equality, but practically, since we derived values directly matching given conditions in the positive set without any inconsistencies.
Thus, it suffices to state the singular appropriate solution available without contradicting complexes or inconsistencies formed outside provided terms:
[tex]\[
(\boxed{7, 6})
\][/tex]
