Write the equation of a line.perpendicular to \( y=5 / 7 x+3 \) and through the point (10,- type your answer...

To find the equation of a line that is perpendicular to the line given by the equation \( y = \frac{5}{7}x + 3 \) and passes through the point \( (10, -y) \), we first need to determine the slope of the original line. The slope of the line \( y = \frac{5}{7}x + 3 \) is \( \frac{5}{7} \). The slope of a line that is perpendicular to this line is the negative reciprocal of \( \frac{5}{7} \). Therefore, the slope \( m \) of the perpendicular line is: \[ m = -\frac{1}{\left(\frac{5}{7}\right)} = -\frac{7}{5} \] Next, we will 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 through which the line passes. In this case, we have the point \( (10, -y) \). We can denote \( y_1 = -y \) and \( x_1 = 10 \). Substituting the values into the point-slope form, we get: \[ y - (-y) = -\frac{7}{5}(x - 10) \] This simplifies to: \[ y + y = -\frac{7}{5}(x - 10) \] Combining the \( y \) terms gives: \[ 2y = -\frac{7}{5}(x - 10) \] Now, we can distribute the right side: \[ 2y = -\frac{7}{5}x + \frac{70}{5} \] This simplifies to: \[ 2y = -\frac{7}{5}x + 14 \] Finally, we can divide everything by 2 to solve for \( y \): \[ y = -\frac{7}{10}x + 7 \] Thus, the equation of the line that is perpendicular to \( y = \frac{5}{7}x + 3 \) and passes through the point \( (10, -y) \) is: \[ y = -\frac{7}{10}x + 7 \]