To solve the given system of equations, we need to determine if there is a common point (x, y) where the two lines intersect. The system of equations is:
[tex]\[ y = \frac{1}{2}x + 2 \][/tex]
[tex]\[ y = -\frac{1}{2}x + 4 \][/tex]
Step-by-Step Solution:
1. Equate the Two Equations:
To find the intersection point, set the equations equal to each other:
[tex]\[ \frac{1}{2}x + 2 = -\frac{1}{2}x + 4 \][/tex]
2. Combine Like Terms:
Add [tex]\(\frac{1}{2}x\)[/tex] to both sides to combine the x terms:
[tex]\[ \frac{1}{2}x + \frac{1}{2}x + 2 = 4 \][/tex]
3. Simplify the Equation:
This simplifies to:
[tex]\[ x + 2 = 4 \][/tex]
4. Solve for x:
Subtract 2 from both sides to isolate x:
[tex]\[ x = 2 \][/tex]
5. Substitute x Back into One of the Original Equations:
Use the first equation to find y:
[tex]\[ y = \frac{1}{2}(2) + 2 \][/tex]
[tex]\[ y = 1 + 2 \][/tex]
[tex]\[ y = 3 \][/tex]
So, the intersection point of the two lines is [tex]\((2, 3)\)[/tex].
Conclusion:
The solution to the system of equations is the single point where the lines intersect, which is [tex]\((2, 3)\)[/tex].
Therefore, the correct answer is: one solution: \{2,3\}
