41) Une \( r \) has an equation of \( y+10=-2(x-6) \). Une \( s \) is perpendicular to line \( r \) and passes through \( (1,2) \). What is the equavon of line \( s \) ?

To find the equation of line \( s \) that is perpendicular to line \( r \) and passes through the point \( (1,2) \), we need to follow these steps: 1. Find the slope of line \( r \) from the given equation. 2. Determine the slope of line \( s \) as the negative reciprocal of the slope of line \( r \). 3. Use the point-slope form of a linear equation to find the equation of line \( s \) that passes through the point \( (1,2) \). Let's start by finding the slope of line \( r \) from the given equation \( y+10=-2(x-6) \). 1. Rearrange the equation to the slope-intercept form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. \[ y+10=-2(x-6) \] \[ y+10=-2x+12 \] \[ y=-2x+2 \] From the equation, we can see that the slope of line \( r \) is -2. 2. The slope of line \( s \) is the negative reciprocal of the slope of line \( r \). Therefore, the slope of line \( s \) is \( \frac{1}{2} \). 3. Now, we can use the point-slope form of a linear equation to find the equation of line \( s \) that passes through the point \( (1,2) \). The point-slope form is given by: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the given point and \( m \) is the slope. Substitute the values into the point-slope form: \[ y - 2 = \frac{1}{2}(x - 1) \] Simplify the equation to find the equation of line \( s \). Solve the equation \( y - 2 = \frac{1}{2}(x - 1) \). Solve the equation by following steps: - step0: Solve for \(y\): \(y-2=\frac{1}{2}\left(x-1\right)\) - step1: Multiply the terms: \(y-2=\frac{1}{2}x-\frac{1}{2}\) - step2: Move the constant to the right side: \(y=\frac{1}{2}x-\frac{1}{2}+2\) - step3: Calculate: \(y=\frac{x+3}{2}\) The equation of line \( s \) is \( y = \frac{x+3}{2} \).