Question

solve the system by the method of your choice.\

$$\begin{cases}6(x - 6) - 5y = 0\\\\5(x - y) = -x - 6\\end{cases}$$

\select the correct choice below and, if necessary, fill in the answer box to complete your choice.\\(\bigcirc\\) a. there is exactly one solution. the solution set is \\(\\{\quad\\}\\).\\(\quad\\) (simplify your answer. type an ordered pair.)\\(\bigcirc\\) b. there are infinitely many solutions. the solution set is \\(\\{(x,y)|6(x - 6) - 5y = 0\\}\\) or \\(\\{(x,y)|5(x - y) = -x - 6\\}\\).\\(\bigcirc\\) c. the solution set is \\(\varnothing\\).

Explanation:

Step1: Simplify the first equation

Start with \( 6(x - 6) - 5y = 0 \).
Expand the left - hand side: \( 6x-36 - 5y=0 \).
Rearrange it to the standard form \( 6x-5y=36 \). Let's call this Equation (1).

Step2: Simplify the second equation

Start with \( 5(x - y)=-x - 6 \).
Expand the left - hand side: \( 5x-5y=-x - 6 \).
Add \( x \) to both sides: \( 5x + x-5y=-6 \).
Combine like terms: \( 6x-5y=-6 \). Let's call this Equation (2).

Step3: Analyze the two equations

We have Equation (1): \( 6x - 5y=36 \) and Equation (2): \( 6x - 5y=-6 \).
Let \( A = 6x-5y \). Then Equation (1) says \( A = 36 \) and Equation (2) says \( A=-6 \). But a single expression \( A \) cannot be equal to two different values at the same time. This means that the system of equations has no solution.

Answer:

C. The solution set is \(\varnothing\).