To solve the given system of linear equations, we need to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations simultaneously. The system of equations is:
[tex]\[ y = -2x + 19 \][/tex]
[tex]\[ y = x + 7 \][/tex]
Here's a step-by-step solution to solve the system of equations:
1. Set the two equations equal to each other:
Since both equations equal [tex]\( y \)[/tex], we set the right-hand sides of the equations equal to each other:
[tex]\[ -2x + 19 = x + 7 \][/tex]
2. Combine like terms:
To isolate [tex]\( x \)[/tex], first combine all [tex]\( x \)[/tex]-terms on one side and constant terms on the other side. Start by subtracting [tex]\( x \)[/tex] from both sides:
[tex]\[ -2x - x + 19 = 7 \][/tex]
[tex]\[ -3x + 19 = 7 \][/tex]
3. Isolate the [tex]\( x \)[/tex]-term:
Subtract 19 from both sides to get:
[tex]\[ -3x = 7 - 19 \][/tex]
[tex]\[ -3x = -12 \][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides by [tex]\(-3\)[/tex]:
[tex]\[ x = \frac{-12}{-3} \][/tex]
[tex]\[ x = 4 \][/tex]
5. Substitute [tex]\( x = 4 \)[/tex] back into one of the original equations:
Let's use the second equation [tex]\( y = x + 7 \)[/tex]:
[tex]\[ y = 4 + 7 \][/tex]
[tex]\[ y = 11 \][/tex]
So, the solution to the system of equations is [tex]\( x = 4 \)[/tex] and [tex]\( y = 11 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{A. \, x = 4, y = 11} \][/tex]
