Answer :
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]
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]