What is the equation of the straight line passing through the point (2,3) and making an intercept on the positive y-axis equal to twice its intercept on the positive x-axis ?

Options

A. `2x + y = 5`
B. `2x + y = 7`
C. `2x - y = 7`
D. `2x - y = 1`

Answer

To find the equation of the straight line that passes through the point (2, 3) and has a y-intercept that is twice its x-intercept, we can follow these steps:

### Step 1: Define the intercepts
Let the x-intercept be \( a \) and the y-intercept be \( b \). According to the problem, we have:
\[ b = 2a \]

### Step 2: Write the equation of the line
The equation of a line in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]

### Step 3: Substitute the intercept relationship into the equation
Substituting \( b = 2a \) into the equation gives:
\[ \frac{x}{a} + \frac{y}{2a} = 1 \]

### Step 4: Multiply through by \( 2a \) to eliminate the denominators
Multiplying the entire equation by \( 2a \) results in:
\[ 2x + y = 2a \]

### Step 5: Substitute the point (2, 3) into the equation
Since the line passes through the point (2, 3), we can substitute \( x = 2 \) and \( y = 3 \) into the equation:
\[ 2(2) + 3 = 2a \]
This simplifies to:
\[ 4 + 3 = 2a \]
\[ 7 = 2a \]

### Step 6: Solve for \( a \)
From the equation \( 7 = 2a \), we can solve for \( a \):
\[ a = \frac{7}{2} \]

### Step 7: Find \( b \)
Using the relationship \( b = 2a \):
\[ b = 2 \left(\frac{7}{2}\right) = 7 \]

### Step 8: Write the final equation of the line
Now that we have both intercepts \( a \) and \( b \), we can substitute back into the intercept form:
\[ \frac{x}{\frac{7}{2}} + \frac{y}{7} = 1 \]

### Step 9: Simplify the equation
Multiplying through by \( 14 \) (the least common multiple of the denominators) gives:
\[ 14 \cdot \frac{x}{\frac{7}{2}} + 14 \cdot \frac{y}{7} = 14 \]
This simplifies to:
\[ 4x + 2y = 14 \]

### Step 10: Rearranging the equation
To get the equation in standard form, we can divide the entire equation by 2:
\[ 2x + y = 7 \]

### Final Answer
Thus, the equation of the straight line is:
\[ 2x + y = 7 \]

---