Question: Write an equation of the line passing through the given point and satisfying the given condition…

Write an equation of the line passing through the given point and satisfying the given condition. Give the equation in slope-intercept form.

(-8, -8); parallel to \(-x + 8y = 24\)

(a) Write the equation of the line in slope-intercept form.

(Simplify your answer. Use integers or fractions for any numbers in the equation.)

(b) Write the equation of the line in standard form.

(Simplify your answer. Use integers or fractions for any numbers in the equation.)

Solution

Write an equation of the line passing through the given point and satisfying the given condition. The point is \((-8, -8)\), and the line is parallel to \(-x + 8y = 24\). To find the equation of the line parallel to the given line in slope-intercept form \(y = mx + b\), we first need the slope of the given line. The equation \(-x + 8y = 24\) can be rewritten in slope-intercept form: First, solve for \(y\): \[ 8y = x + 24 \] Divide by 8: \[ y = \frac{1}{8}x + 3 \] The slope \(m\) of the line is \(\frac{1}{8}\). Since parallel lines have the same slope, the slope of the desired line is also \(\frac{1}{8}\). Use the point-slope form of the line equation, \(y - y_1 = m(x - x_1)\), with the point \((-8, -8)\) and slope \(\frac{1}{8}\): \[ y + 8 = \frac{1}{8}(x + 8) \] Distribute the \(\frac{1}{8}\): \[ y + 8 = \frac{1}{8}x + 1 \] Subtract 8 from both sides to get the slope-intercept form: \[ y = \frac{1}{8}x - 7 \] Now, convert this to the standard form \(Ax + By = C\): Multiply every term by 8 to eliminate the fraction: \[ 8y = x - 56 \] Rearrange to get the standard form: \[ -x + 8y = -56 \] (a) The equation of the line in slope-intercept form is: \[ y = \frac{1}{8}x - 7 \] (b) The equation of the line in standard form is: \[ -x + 8y = -56 \]