To find the tangent of the acute angle between the two lines given by the equations \(2x + y = 3\) and \(3x - 2y = 5\), we will follow a systematic approach. ### Step 1: Find the slopes of the lines First, we need to rewrite both equations in the slope-intercept form \(y = mx + b\), where \(m\) is the slope. **For the first line \(2x + y = 3\):** 1. Rearranging gives: \[ y = -2x + 3 \] Here, the slope \(m_1 = -2\). **For the second line \(3x - 2y = 5\):** 1. Rearranging gives: \[ -2y = -3x + 5 \quad \Rightarrow \quad y = \frac{3}{2}x - \frac{5}{2} \] Here, the slope \(m_2 = \frac{3}{2}\). ### Step 2: Use the formula for the tangent of the angle between two lines The formula for the tangent of the angle \(\theta\) between two lines with slopes \(m_1\) and \(m_2\) is given by: \[ \tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right| \] ### Step 3: Substitute the slopes into the formula Now we substitute \(m_1 = -2\) and \(m_2 = \frac{3}{2}\) into the formula: 1. Calculate \(m_1 - m_2\): \[ m_1 - m_2 = -2 - \frac{3}{2} = -\frac{4}{2} - \frac{3}{2} = -\frac{7}{2} \] 2. Calculate \(1 + m_1 m_2\): \[ m_1 m_2 = -2 \cdot \frac{3}{2} = -3 \quad \Rightarrow \quad 1 + m_1 m_2 = 1 - 3 = -2 \] ### Step 4: Substitute into the tangent formula Now we substitute these values into the tangent formula: \[ \tan(\theta) = \left| \frac{-\frac{7}{2}}{-2} \right| = \left| \frac{7}{2} \cdot \frac{1}{-2} \right| = \left| \frac{7}{-4} \right| = \frac{7}{4} \] ### Conclusion: Identify the correct option The tangent of the acute angle between the two lines is \(\frac{7}{4}\). Therefore, the correct option is: **C. 7/4** ### Explanation of Other Options - **Option A: -7/4** - This is incorrect because the tangent of an angle cannot be negative when considering the acute angle between two lines. - **Option B: 7/8** - This is incorrect as it does not match our calculated value of \(\frac{7}{4}\). - **Option D: 7/2** - This is also incorrect as it is not the result of our calculations. ### Revision Summary - To find the tangent of the angle between two lines, first determine their slopes. - Use the formula \(\tan(\theta) = \left| \frac{m_1 - m_2}{1 + m_1 m_2} \right|\). - Substitute the slopes into the formula to find the tangent value. - The correct answer for the tangent of the acute angle between the lines is \(\frac{7}{4}\).