What is the height of triangle formed by straight lines `3x + y = 10, 2x - 3y = 6` and `x`-axis when `x`-axis is the base of the triangle ?

Options

A. 3
B. 1
C. `5/2`
D. `7/2`

Answer

To find the height of the triangle formed by the lines \(3x + y = 10\), \(2x - 3y = 6\), and the x-axis, we can follow these steps:

### Step 1: Find the intersection point of the two lines
We need to solve the equations \(3x + y = 10\) and \(2x - 3y = 6\) simultaneously to find the coordinates of the intersection point, which we will call point A.

1. Rearranging the first equation:
\[ y = 10 - 3x \quad \text{(Equation 1)} \]

2. Substitute \(y\) from Equation 1 into the second equation:
\[ 2x - 3(10 - 3x) = 6 \]
\[ 2x - 30 + 9x = 6 \]
\[ 11x - 30 = 6 \]
\[ 11x = 36 \]
\[ x = \frac{36}{11} \]

3. Now substitute \(x\) back into Equation 1 to find \(y\):
\[ y = 10 - 3\left(\frac{36}{11}\right) \]
\[ y = 10 - \frac{108}{11} \]
\[ y = \frac{110}{11} - \frac{108}{11} = \frac{2}{11} \]

Thus, the coordinates of point A are:
\[ A\left(\frac{36}{11}, \frac{2}{11}\right) \]

### Step 2: Determine the height of the triangle
The height of the triangle formed by the lines and the x-axis is simply the y-coordinate of point A, as the x-axis serves as the base.

Thus, the height \(h\) is:
\[ h = \frac{2}{11} \]

### Final Answer
The height of the triangle formed by the lines \(3x + y = 10\), \(2x - 3y = 6\), and the x-axis is:
\[ \frac{2}{11} \]

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