The straight line `2x +3y = 12` passes through :

Options

A. 1 st, 2nd and 3rd quadrant
B. 1st , 2nd and 4th quadrant
C. 2 nd, 3rd and 4 th quadrant
D. 1 st, 3rd and 4th quadrant

Answer

To determine through which quadrants the straight line represented by the equation \(2x + 3y = 12\) passes, we can follow these steps:

### Step 1: Find the x-intercept
To find the x-intercept, we set \(y = 0\) in the equation:

\[ 2x + 3(0) = 12 \implies 2x = 12 \implies x = \frac{12}{2} = 6 \]

Thus, the x-intercept is at the point \((6, 0)\).

### Step 2: Find the y-intercept
Next, we find the y-intercept by setting \(x = 0\) in the equation:

\[ 2(0) + 3y = 12 \implies 3y = 12 \implies y = \frac{12}{3} = 4 \]

Thus, the y-intercept is at the point \((0, 4)\).

### Step 3: Plot the intercepts
We now plot the points \((6, 0)\) and \((0, 4)\) on a coordinate plane. The point \((6, 0)\) lies on the positive x-axis, and the point \((0, 4)\) lies on the positive y-axis.

### Step 4: Draw the line
By connecting the two points \((6, 0)\) and \((0, 4)\), we can visualize the line represented by the equation \(2x + 3y = 12\).

### Step 5: Determine the quadrants
Now we need to analyze the line's position relative to the quadrants:
- The line starts from the y-intercept \((0, 4)\) and moves downwards to the x-intercept \((6, 0)\).
- Since the line passes through the first quadrant (where both x and y are positive) and extends into the fourth quadrant (where x is positive and y is negative), we can conclude that the line also crosses into the third quadrant (where both x and y are negative).

### Conclusion
The line \(2x + 3y = 12\) passes through the first, fourth, and third quadrants.