Answer :
Sure! Let's solve the given system of equations step by step to find the correct solution.
The system of equations provided is:
[tex]\[ \begin{cases} 2x + 4y = 12 \\ y = \frac{1}{4}x - 3 \end{cases} \][/tex]
### Step 1: Substitute [tex]\( y \)[/tex] from the second equation into the first equation.
The second equation is [tex]\( y = \frac{1}{4}x - 3 \)[/tex].
We substitute [tex]\( y \)[/tex] in the first equation [tex]\( 2x + 4y = 12 \)[/tex]:
[tex]\[ 2x + 4\left(\frac{1}{4}x - 3\right) = 12 \][/tex]
### Step 2: Simplify the substituted equation.
First, distribute the 4 inside the parentheses:
[tex]\[ 2x + 4 \cdot \left(\frac{1}{4}x\right) - 4 \cdot 3 = 12 \][/tex]
This simplifies to:
[tex]\[ 2x + x - 12 = 12 \][/tex]
Combine like terms:
[tex]\[ 3x - 12 = 12 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex].
Add 12 to both sides of the equation to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 3x = 24 \][/tex]
Divide both sides by 3:
[tex]\[ x = 8 \][/tex]
### Step 4: Substitute [tex]\( x \)[/tex] back into the second equation to solve for [tex]\( y \)[/tex].
The second equation is [tex]\( y = \frac{1}{4}x - 3 \)[/tex]:
[tex]\[ y = \frac{1}{4} \cdot 8 - 3 \][/tex]
Simplify the right side:
[tex]\[ y = 2 - 3 \][/tex]
[tex]\[ y = -1 \][/tex]
### Conclusion
The solution to the system of equations is [tex]\( (8, -1) \)[/tex]. Thus, the correct choice among the given options is:
[tex]\[ (8, -1) \][/tex]
The system of equations provided is:
[tex]\[ \begin{cases} 2x + 4y = 12 \\ y = \frac{1}{4}x - 3 \end{cases} \][/tex]
### Step 1: Substitute [tex]\( y \)[/tex] from the second equation into the first equation.
The second equation is [tex]\( y = \frac{1}{4}x - 3 \)[/tex].
We substitute [tex]\( y \)[/tex] in the first equation [tex]\( 2x + 4y = 12 \)[/tex]:
[tex]\[ 2x + 4\left(\frac{1}{4}x - 3\right) = 12 \][/tex]
### Step 2: Simplify the substituted equation.
First, distribute the 4 inside the parentheses:
[tex]\[ 2x + 4 \cdot \left(\frac{1}{4}x\right) - 4 \cdot 3 = 12 \][/tex]
This simplifies to:
[tex]\[ 2x + x - 12 = 12 \][/tex]
Combine like terms:
[tex]\[ 3x - 12 = 12 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex].
Add 12 to both sides of the equation to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 3x = 24 \][/tex]
Divide both sides by 3:
[tex]\[ x = 8 \][/tex]
### Step 4: Substitute [tex]\( x \)[/tex] back into the second equation to solve for [tex]\( y \)[/tex].
The second equation is [tex]\( y = \frac{1}{4}x - 3 \)[/tex]:
[tex]\[ y = \frac{1}{4} \cdot 8 - 3 \][/tex]
Simplify the right side:
[tex]\[ y = 2 - 3 \][/tex]
[tex]\[ y = -1 \][/tex]
### Conclusion
The solution to the system of equations is [tex]\( (8, -1) \)[/tex]. Thus, the correct choice among the given options is:
[tex]\[ (8, -1) \][/tex]
