Questions: Problem Find the equation of the line passing through the point (6,3) that is perpendicular to the line 4 x-5 y=-10. Enter your answers below. Use a forward slash (i.e. "/") for fractions (e.g. 1 / 2 for 1/2 ). Solution Step 1: Find the slope of the line 4 x-5 y=-10. Use a forward slash (i.e. "/") for all fractions (e.g. 1 / 2 for 1/2 ). m= What would the perpendicular slope be? m= Step 2: Use the slope to find the y-intercept of the perpendicular line. b= Step 3: Write the equation of the line that passes through the point (6,3) that is perpendicular to the line 4 x-5 y=-10 y= x+

Solution Steps

1. First, convert the given line equation \(4x - 5y = -10\) into the slope-intercept form \(y = mx + b\) to find its slope.
2. The slope of the line perpendicular to the given line is the negative reciprocal of the original slope.
3. Use the point-slope form of the line equation with the perpendicular slope and the given point \((6, 3)\) to find the equation of the new line.

The equation of the line is given by \(4x - 5y = -10\). To find the slope, we can rearrange this equation into the slope-intercept form \(y = mx + b\). Solving for \(y\), we get: \[ y = \frac{4}{5}x + 2 \] Thus, the slope of the original line is \(m = \frac{4}{5}\).

The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, the perpendicular slope is: \[ m_{\text{perpendicular}} = -\frac{5}{4} \]

Using the point-slope form of the line equation, which is \(y - y_1 = m(x - x_1)\), where \((x_1, y_1) = (6, 3)\) and \(m = -\frac{5}{4}\), we can write: \[ y - 3 = -\frac{5}{4}(x - 6) \] Expanding this, we find: \[ y - 3 = -\frac{5}{4}x + \frac{30}{4} \] Simplifying gives: \[ y = -\frac{5}{4}x + \frac{21}{2} \]

Final Answer