Answer :
To solve the system of equations
[tex]\[ \begin{cases} -8x - 8y = 0 \\ -8x + 2y = -20 \end{cases} \][/tex]
we can use the method of substitution or elimination. Here, I'll demonstrate both the methods for clarity, and we'll check which answer is correct among the given options.
### Method 1: Substitution Method
1. Let's take the first equation:
[tex]\[ -8x - 8y = 0 \][/tex]
2. From this equation, we can solve for [tex]\( y \)[/tex]:
[tex]\[ -8x = 8y \\ y = -x \][/tex]
3. Substitute [tex]\( y = -x \)[/tex] into the second equation:
[tex]\[ -8x + 2(-x) = -20 \\ -8x - 2x = -20 \\ -10x = -20 \\ x = 2 \][/tex]
4. Now, substitute [tex]\( x = 2 \)[/tex] back into [tex]\( y = -x \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[ y = -2 \][/tex]
Thus, the solution to the system is [tex]\( (2, -2) \)[/tex].
### Method 2: Elimination Method
1. Write down the system of equations:
[tex]\[ \begin{cases} -8x - 8y = 0 \\ -8x + 2y = -20 \end{cases} \][/tex]
2. To eliminate [tex]\( x \)[/tex], we can subtract the first equation from the second equation:
[tex]\[ \big( -8x + 2y \big) - \big( -8x - 8y \big) = -20 - 0 \\ -8x + 2y + 8x + 8y = -20 \\ 10y = -20 \\ y = -2 \][/tex]
3. Substitute [tex]\( y = -2 \)[/tex] back into the first equation to find [tex]\( x \)[/tex]:
[tex]\[ -8x - 8(-2) = 0 \\ -8x + 16 = 0 \\ -8x = -16 \\ x = 2 \][/tex]
Thus, the solution to the system is [tex]\( (2, -2) \)[/tex].
### Conclusion
The solution to the system of equations
[tex]\[ \begin{cases} -8x - 8y = 0 \\ -8x + 2y = -20 \end{cases} \][/tex]
is [tex]\((2, -2)\)[/tex].
Therefore, the correct answer is:
A. [tex]\((2, -2)\)[/tex]
[tex]\[ \begin{cases} -8x - 8y = 0 \\ -8x + 2y = -20 \end{cases} \][/tex]
we can use the method of substitution or elimination. Here, I'll demonstrate both the methods for clarity, and we'll check which answer is correct among the given options.
### Method 1: Substitution Method
1. Let's take the first equation:
[tex]\[ -8x - 8y = 0 \][/tex]
2. From this equation, we can solve for [tex]\( y \)[/tex]:
[tex]\[ -8x = 8y \\ y = -x \][/tex]
3. Substitute [tex]\( y = -x \)[/tex] into the second equation:
[tex]\[ -8x + 2(-x) = -20 \\ -8x - 2x = -20 \\ -10x = -20 \\ x = 2 \][/tex]
4. Now, substitute [tex]\( x = 2 \)[/tex] back into [tex]\( y = -x \)[/tex] to find [tex]\( y \)[/tex]:
[tex]\[ y = -2 \][/tex]
Thus, the solution to the system is [tex]\( (2, -2) \)[/tex].
### Method 2: Elimination Method
1. Write down the system of equations:
[tex]\[ \begin{cases} -8x - 8y = 0 \\ -8x + 2y = -20 \end{cases} \][/tex]
2. To eliminate [tex]\( x \)[/tex], we can subtract the first equation from the second equation:
[tex]\[ \big( -8x + 2y \big) - \big( -8x - 8y \big) = -20 - 0 \\ -8x + 2y + 8x + 8y = -20 \\ 10y = -20 \\ y = -2 \][/tex]
3. Substitute [tex]\( y = -2 \)[/tex] back into the first equation to find [tex]\( x \)[/tex]:
[tex]\[ -8x - 8(-2) = 0 \\ -8x + 16 = 0 \\ -8x = -16 \\ x = 2 \][/tex]
Thus, the solution to the system is [tex]\( (2, -2) \)[/tex].
### Conclusion
The solution to the system of equations
[tex]\[ \begin{cases} -8x - 8y = 0 \\ -8x + 2y = -20 \end{cases} \][/tex]
is [tex]\((2, -2)\)[/tex].
Therefore, the correct answer is:
A. [tex]\((2, -2)\)[/tex]
