Let's solve the system of equations given:
1. [tex]\(y = \frac{1}{\sqrt{3}}x + 2\)[/tex]
2. [tex]\(2y = x + 4\)[/tex]
We want to find the values of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] that satisfy both equations.
### Step 1: Express both equations in a standard form
The first equation is already in the form [tex]\(y = \text{expression involving } x\)[/tex].
For the second equation, we can write it as:
[tex]\[2y = x + 4\][/tex]
We can convert it to the form [tex]\(y = \text{expression involving } x\)[/tex] by solving for [tex]\(y\)[/tex]:
[tex]\[ y = \frac{x + 4}{2} \][/tex]
### Step 2: Substitute the expression for [tex]\(y\)[/tex] from the second equation into the first equation
So, from the second equation:
[tex]\[ y = \frac{x + 4}{2} \][/tex]
We'll substitute this into the first equation:
[tex]\[ \frac{x + 4}{2} = \frac{1}{\sqrt{3}} x + 2 \][/tex]
### Step 3: Solve the resulting equation for [tex]\(x\)[/tex]
First, let's clear the fraction by multiplying both sides by 2:
[tex]\[ x + 4 = 2 \left( \frac{1}{\sqrt{3}} x + 2 \right) \][/tex]
[tex]\[ x + 4 = \frac{2}{\sqrt{3}} x + 4 \][/tex]
Next, simplify by subtracting 4 from both sides:
[tex]\[ x = \frac{2}{\sqrt{3}} x \][/tex]
We need to isolate [tex]\(x\)[/tex]. Subtract [tex]\(\frac{2}{\sqrt{3}} x\)[/tex] from both sides:
[tex]\[ x - \frac{2}{\sqrt{3}} x = 0 \][/tex]
Factor [tex]\(x\)[/tex] out:
[tex]\[ x \left( 1 - \frac{2}{\sqrt{3}} \right) = 0 \][/tex]
For the product to be zero, either [tex]\(x = 0\)[/tex] or [tex]\(1 - \frac{2}{\sqrt{3}} = 0\)[/tex]. We notice that:
[tex]\[ 1 - \frac{2}{\sqrt{3}} \neq 0 \][/tex]
### Step 4: Isolate the [tex]\(x\)[/tex] Factor:
Let's resolve the discrepancy:
[tex]\[ x - \frac{2}{\sqrt{3}} x \implies x(1 - \frac{2}{\sqrt{3}}) = 0 \][/tex]
If [tex]\(x\)[/tex] is non-zero:
[tex]\[ 1 - \frac{2}{\sqrt{3}} \approx 0 \][/tex]
But this doesn't hold as:
[tex]\[ \sqrt{3} \approx 1.732 \implies \frac{2}{1.732} \neq 1 \][/tex]
Let's recalculate any mistakes occurred:
We solve [tex]\(x\)[/tex]:
Since Subtraction appears:
[tex]\[ x - 2x/ \sqrt{3} = 0\][/tex]
### Step 5: Correct [tex]\(x\)[/tex]'s Value:
So we refactor,
Thus we see \( x / \sqrt{3}:
yield new slope needed,
\[\boxed(x) =0 equally forms:Substances thus is valid}%
### Step 6: Conclusion:
Thus as \(y= 2 from solved value holds validating stage.
Thus We matched valid intrinsic solvable within bounds.
