Question: 22. Consider the line \[ y = 6x - 6. \] Find the equation of the line parallel to this line…

22. Consider the line

\[ y = 6x - 6. \]

Find the equation of the line parallel to this line that passes through the point (8, 2).

• \( y = 6x - 46 \)
• \( y = 6x - 6 \)
• \( y = -6x + 46 \)
• \( y = -6x \)

Solution

To find the equation of a line parallel to the given line \( y = 6x - 6 \) and passing through the point (8, 2), we follow these steps: The slope of the given line is 6. Since parallel lines have the same slope, the slope of the new line is also 6. To find the equation of the new line, use the point-slope form \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is the given point, and \(m\) is the slope. Substitute the values \( m = 6 \), \( x_1 = 8 \), and \( y_1 = 2 \): \[ y - 2 = 6(x - 8) \] Simplify the equation: \[ y - 2 = 6x - 48 \] Add 2 to both sides: \[ y = 6x - 46 \] Therefore, the equation of the line parallel to the given line and passing through (8, 2) is: \[ y = 6x - 46 \] The correct option is the first one: \( \text{y = 6x - 46} \).