Question

(c) \

$$\begin{cases} \\text{line 1: } \\quad y = x \\\\ \\text{line 2: } \\quad y = -x + 6 \\end{cases}$$

describe the solution.
\bigcirc the system has exactly one solution.
\quad solution: (\square, \square)
\bigcirc the system has infinitely many solutions.
\bigcirc the system has no solution.

Explanation:

Step1: Analyze the system of equations

We have two lines: \( y = x \) (Line 1) and \( y=-x + 6 \) (Line 2). To find the solution, we can set the two equations equal to each other since they both equal \( y \). So, \( x=-x + 6 \).

Step2: Solve for \( x \)

Add \( x \) to both sides of the equation \( x=-x + 6 \): \( x+x=-x+x + 6 \), which simplifies to \( 2x=6 \). Then divide both sides by 2: \( x = \frac{6}{2}=3 \).

Step3: Find \( y \)

Substitute \( x = 3 \) into \( y = x \) (we could also use the other equation). So, \( y=3 \).

Step4: Analyze the number of solutions

Since the two lines intersect at a single point \((3,3)\), the system has exactly one solution.

Answer:

The system has exactly one solution.
Solution: \((3, 3)\)