The graphs of the equations 3x + y - 5 = 0 and 2x - y = 0 intersect at the point P`(alpha,beta)` . What is the value of `(3alpha + beta)` ?

Options

A. 4
B. -4
C. 5
D. 3

Answer

To solve the problem, we need to find the point of intersection of the two equations given and then calculate the value of \(3\alpha + \beta\).

### Step 1: Write down the equations
The equations given are:
1. \(3x + y - 5 = 0\)
2. \(2x - y = 0\)

### Step 2: Solve the equations simultaneously
We can solve these equations by substitution or elimination. Here, we will use the elimination method.

First, we can rearrange the second equation to express \(y\) in terms of \(x\):
\[ y = 2x \]

### Step 3: Substitute \(y\) in the first equation
Now we substitute \(y = 2x\) into the first equation:
\[ 3x + (2x) - 5 = 0 \]
This simplifies to:
\[ 5x - 5 = 0 \]

### Step 4: Solve for \(x\)
Now, we can solve for \(x\):
\[ 5x = 5 \implies x = 1 \]

### Step 5: Find \(y\) using the value of \(x\)
Now that we have \(x\), we can find \(y\) using the equation \(y = 2x\):
\[ y = 2(1) = 2 \]

### Step 6: Identify the point of intersection
The point of intersection \(P(\alpha, \beta)\) is:
\[ P(1, 2) \]
Thus, \(\alpha = 1\) and \(\beta = 2\).

### Step 7: Calculate \(3\alpha + \beta\)
Now we need to calculate \(3\alpha + \beta\):
\[ 3\alpha + \beta = 3(1) + 2 = 3 + 2 = 5 \]

### Final Answer
The value of \(3\alpha + \beta\) is \(5\).

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