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Solve the system of equations:

[tex]\[
\begin{array}{l}
2x + 4y = 12 \\
y = \frac{1}{4}x - 3
\end{array}
\][/tex]

What is the solution to the system of equations?

A. [tex]$(-1, 8)$[/tex]
B. [tex]$(8, -1)$[/tex]
C. [tex]$\left(5, \frac{1}{2}\right)$[/tex]
D. [tex]$\left(\frac{1}{2}, 5\right)$[/tex]


Sagot :

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]