h) Find the equation of the line that is perpendicular to the line \( -2 x+5 y=10 \) and passes through the point \( (2,3) \). Write answer in slope-intercept form.

To find the equation of the line that is perpendicular to the line given by the equation \(-2x + 5y = 10\) and passes through the point \((2, 3)\), we can follow these steps: 1. **Find the slope of the given line**: We first need to rewrite the equation in slope-intercept form \(y = mx + b\), where \(m\) is the slope. Starting with the equation: \[ -2x + 5y = 10 \] We can solve for \(y\): \[ 5y = 2x + 10 \] \[ y = \frac{2}{5}x + 2 \] From this, we see that the slope \(m\) of the given line is \(\frac{2}{5}\). 2. **Find the slope of the perpendicular line**: The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Therefore, the slope \(m_{\perp}\) of the perpendicular line is: \[ m_{\perp} = -\frac{1}{\frac{2}{5}} = -\frac{5}{2} \] 3. **Use the point-slope form to find the equation of the perpendicular line**: We can use the point-slope form of the equation of a line, which is given by: \[ y - y_1 = m(x - x_1) \] where \((x_1, y_1)\) is the point the line passes through, and \(m\) is the slope. Here, \((x_1, y_1) = (2, 3)\) and \(m = -\frac{5}{2}\). Plugging in these values: \[ y - 3 = -\frac{5}{2}(x - 2) \] 4. **Simplify to slope-intercept form**: Distributing the slope on the right side: \[ y - 3 = -\frac{5}{2}x + 5 \] Now, add 3 to both sides: \[ y = -\frac{5}{2}x + 5 + 3 \] \[ y = -\frac{5}{2}x + 8 \] Thus, the equation of the line that is perpendicular to the line \(-2x + 5y = 10\) and passes through the point \((2, 3)\) in slope-intercept form is: \[ \boxed{y = -\frac{5}{2}x + 8} \]