To determine the value of \( K \) such that the lines represented by the equations \( 3y = 4x - 1 \) and \( Ky = x + 3 \) are perpendicular, we need to follow these steps: ### Step 1: Convert the equations to slope-intercept form The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope of the line. **For the first equation:** \[ 3y = 4x - 1 \] To convert this to slope-intercept form, divide every term by 3: \[ y = \frac{4}{3}x - \frac{1}{3} \] From this, we can see that the slope \( m_1 \) of the first line is: \[ m_1 = \frac{4}{3} \] **For the second equation:** \[ Ky = x + 3 \] To convert this to slope-intercept form, divide every term by \( K \): \[ y = \frac{1}{K}x + \frac{3}{K} \] From this, we can see that the slope \( m_2 \) of the second line is: \[ m_2 = \frac{1}{K} \] ### Step 2: Use the condition for perpendicular lines Two lines are perpendicular if the product of their slopes is equal to -1. Therefore, we can set up the following equation: \[ m_1 \cdot m_2 = -1 \] Substituting the slopes we found: \[ \left(\frac{4}{3}\right) \cdot \left(\frac{1}{K}\right) = -1 \] ### Step 3: Solve for \( K \) Now, we can solve for \( K \): \[ \frac{4}{3K} = -1 \] To eliminate the fraction, multiply both sides by \( 3K \): \[ 4 = -3K \] Now, divide both sides by -3: \[ K = -\frac{4}{3} \] ### Conclusion Thus, the value of \( K \) that makes the two lines perpendicular is: **A. -4/3** ### Explanation of Other Options - **B. -3/4**: This value does not satisfy the condition for perpendicularity since substituting \( K = -\frac{3}{4} \) into the slope equation would not yield a product of -1. - **C. 3/4**: This value would yield a positive product of slopes, which cannot satisfy the perpendicular condition. - **D. 4/3**: This would also yield a positive product of slopes, which is not suitable for perpendicular lines. ### Revision Summary - Convert line equations to slope-intercept form to identify slopes. - Use the condition \( m_1 \cdot m_2 = -1 \) for perpendicular lines. - Solve for \( K \) to find the required value. - The correct answer is \( K = -\frac{4}{3} \).