To determine the solution to the given system of equations, follow these steps carefully:
Given the system:
[tex]\[
\begin{cases}
x + y = 6 \\
x = y + 4
\end{cases}
\][/tex]
1. Substitute the second equation into the first:
Since [tex]\( x = y + 4 \)[/tex], we can substitute this expression for [tex]\( x \)[/tex] in the first equation.
Substitute [tex]\( x \)[/tex] in [tex]\( x + y = 6 \)[/tex]:
[tex]\[
(y + 4) + y = 6
\][/tex]
2. Simplify and solve for [tex]\( y \)[/tex]:
Combine like terms:
[tex]\[
y + 4 + y = 6
\][/tex]
[tex]\[
2y + 4 = 6
\][/tex]
Isolate [tex]\( y \)[/tex] by first subtracting 4 from both sides:
[tex]\[
2y = 2
\][/tex]
Then, divide by 2:
[tex]\[
y = 1
\][/tex]
3. Substitute [tex]\( y = 1 \)[/tex] back into the second equation to find [tex]\( x \)[/tex]:
Using [tex]\( x = y + 4 \)[/tex]:
[tex]\[
x = 1 + 4
\][/tex]
[tex]\[
x = 5
\][/tex]
4. Write the solution as an ordered pair:
The solution to the system is:
[tex]\[
(x, y) = (5, 1)
\][/tex]
So, the correct solution to the system of equations is [tex]\((5, 1)\)[/tex]. Thus, the answer is:
[tex]\[
(5, 1)
\][/tex]
