A line passes through the point \( (7,10) \) and has a slope of \( \frac{2}{7} \). Write the equation of the line using \( x \) as the independent variable: \( y=\square \) Find the vertical intercept for this line:

To find the equation of the line that passes through the point \( (7, 10) \) with a slope of \( \frac{2}{7} \), 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 a point on the line and \( m \) is the slope. Here, \( (x_1, y_1) = (7, 10) \) and \( m = \frac{2}{7} \). Substituting these values into the point-slope form: \[ y - 10 = \frac{2}{7}(x - 7) \] Now, we can simplify this equation to get it into slope-intercept form \( y = mx + b \). First, distribute the slope on the right side: \[ y - 10 = \frac{2}{7}x - \frac{2}{7} \cdot 7 \] Calculating \( \frac{2}{7} \cdot 7 \): \[ y - 10 = \frac{2}{7}x - 2 \] Next, add 10 to both sides to isolate \( y \): \[ y = \frac{2}{7}x - 2 + 10 \] Simplifying the right side: \[ y = \frac{2}{7}x + 8 \] Thus, the equation of the line is: \[ y = \frac{2}{7}x + 8 \] Now, to find the vertical intercept (also known as the y-intercept), we can set \( x = 0 \) in the equation: \[ y = \frac{2}{7}(0) + 8 = 8 \] Therefore, the vertical intercept is: \[ \text{Vertical intercept: } 8 \] In summary, the equation of the line is: \[ y = \frac{2}{7}x + 8 \] And the vertical intercept is: \[ 8 \]