Alright, let's solve the given system of equations step-by-step to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
The system of equations is:
[tex]\[
\begin{cases}
x - y = 1 \\
x + y = 5
\end{cases}
\][/tex]
Step 1: Add the two equations
First, we add the two equations to eliminate [tex]\( y \)[/tex]:
[tex]\[
(x - y) + (x + y) = 1 + 5
\][/tex]
Simplify the left-hand side:
[tex]\[
x - y + x + y = 1 + 5
\][/tex]
The [tex]\( -y \)[/tex] and [tex]\( +y \)[/tex] terms cancel each other out:
[tex]\[
2x = 6
\][/tex]
Step 2: Solve for [tex]\( x \)[/tex]
Divide both sides of the equation by 2 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{6}{2} = 3
\][/tex]
So, we have [tex]\( x = 3 \)[/tex].
Step 3: Substitute [tex]\( x \)[/tex] back into one of the original equations
Next, we substitute [tex]\( x = 3 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]. Let's use [tex]\( x + y = 5 \)[/tex]:
[tex]\[
3 + y = 5
\][/tex]
Step 4: Solve for [tex]\( y \)[/tex]
Subtract 3 from both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 5 - 3 = 2
\][/tex]
So, we have [tex]\( y = 2 \)[/tex].
Conclusion
The solution to the system of equations is:
[tex]\[
x = 3, \quad y = 2
\][/tex]
