To solve the given system of equations using different approaches, let's break down each option in detail and find the results.
Given the system of equations:
[tex]\[ \begin{cases}
2x + 8y = -3 \\
3x + 6y = -4
\end{cases} \][/tex]
Option A: Multiply the top equation by 3, multiply the bottom equation by 4, then subtract the bottom equation from the top equation.
1. Multiply the top equation by 3:
[tex]\[
3(2x + 8y) = 3(-3)
\][/tex]
[tex]\[
6x + 24y = -9
\][/tex]
2. Multiply the bottom equation by 4:
[tex]\[
4(3x + 6y) = 4(-4)
\][/tex]
[tex]\[
12x + 24y = -16
\][/tex]
3. Subtract the bottom equation from the top equation:
[tex]\[
(6x + 24y) - (12x + 24y) = -9 - (-16)
\][/tex]
[tex]\[
6x + 24y - 12x - 24y = -9 + 16
\][/tex]
[tex]\[
-6x = 7
\][/tex]
This indicates:
[tex]\[
x = -\frac{7}{6}
\][/tex]
So, the first valid operation gives us [tex]\( x = -\frac{7}{6} \)[/tex].
Option B: Multiply the top equation by -4, multiply the bottom equation by 3, then add the equations.
1. Multiply the top equation by -4:
[tex]\[
-4(2x + 8y) = -4(-3)
\][/tex]
[tex]\[
-8x - 32y = 12
\][/tex]
2. Multiply the bottom equation by 3:
[tex]\[
3(3x + 6y) = 3(-4)
\][/tex]
[tex]\[
9x + 18y = -12
\][/tex]
3. Add the equations:
[tex]\[
(-8x - 32y) + (9x + 18y) = 12 + (-12)
\][/tex]
[tex]\[
-8x + 9x - 32y + 18y = 0
\][/tex]
[tex]\[
x - 14y = 0
\][/tex]
This implies:
[tex]\[
x = 14y
\][/tex]
So, the second valid operation reduces the system to [tex]\( x = 14y \)[/tex].
Option C: Multiply the top equation by 3, multiply the bottom equation by -2, then add the equations.
1. Multiply the top equation by 3:
[tex]\[
3(2x + 8y) = 3(-3)
\][/tex]
[tex]\[
6x + 24y = -9
\][/tex]
2. Multiply the bottom equation by -2:
[tex]\[
-2(3x + 6y) = -2(-4)
\][/tex]
[tex]\[
-6x - 12y = 8
\][/tex]
3. Add the equations:
[tex]\[
(6x + 24y) + (-6x - 12y) = -9 + 8
\][/tex]
[tex]\[
6x - 6x + 24y - 12y = -1
\][/tex]
[tex]\[
12y = -1
\][/tex]
This gives us:
[tex]\[
y = -\frac{1}{12}
\][/tex]
So, the third valid operation provides [tex]\( y = -\frac{1}{12} \)[/tex].
To summarize:
- Option A: Results in [tex]\( x = -\frac{7}{6} \)[/tex]
- Option B: Results in [tex]\( x = 14y \)[/tex]
- Option C: Results in [tex]\( y = -\frac{1}{12} \)[/tex]
Both Options A and C are valid logical steps to solve the system of equations. Option B too, but it doesn't directly provide a solution, instead, it shows the relationship between [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
