Questions: Find the equation of the line that contains the point (6, -2) and is perpendicular to the line y = -2x + 8. y = 2x - 14 y = x - 8 y = 1/2x - 5 y = -2x + 10

Solution Steps

To find the equation of a line that is perpendicular to a given line and passes through a specific point, we need to follow these steps:

1. Determine the slope of the given line. The line is in the form \( y = mx + b \), where \( m \) is the slope. For the line \( y = -2x + 8 \), the slope \( m \) is -2.
2. The slope of a line perpendicular to another is the negative reciprocal of the original slope. Therefore, the perpendicular slope will be \( \frac{1}{2} \).
3. Use the point-slope form of a line equation, \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) is the point the line passes through, and \( m \) is the slope. Substitute the point (6, -2) and the perpendicular slope \( \frac{1}{2} \) into this equation.
4. Simplify the equation to get it into the slope-intercept form \( y = mx + b \).

The equation of the given line is \( y = -2x + 8 \). The slope \( m \) of this line is \( -2 \).

The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, the perpendicular slope is: \[ m_{\text{perpendicular}} = -\frac{1}{m} = -\frac{1}{-2} = \frac{1}{2} \]

We will use the point-slope form of the line equation, which is given by: \[ y - y_1 = m(x - x_1) \] Substituting the point \( (6, -2) \) and the perpendicular slope \( \frac{1}{2} \): \[ y - (-2) = \frac{1}{2}(x - 6) \] This simplifies to: \[ y + 2 = \frac{1}{2}x - 3 \]

Now, we rearrange the equation to get it into the slope-intercept form \( y = mx + b \): \[ y = \frac{1}{2}x - 3 - 2 \] \[ y = \frac{1}{2}x - 5 \]

Final Answer