Question: (a) Write the equation of the line in slope-intercept form. (b) Write the equation of the line in…

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

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

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

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

\((-3, -3)\), parallel to \(y = x + 9y\)

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Solution

Write an equation of the line passing through the given point and satisfying the given condition. Give the equation (a) in slope-intercept form and (b) in standard form. Point: \((-8, -8)\) Parallel to the line: \(x + 3y = 24\) 1. First, find the slope of the line \(x + 3y = 24\). Rewrite the equation in slope-intercept form \(y = mx + b\). \[ 3y = -x + 24 \] \[ y = -\frac{1}{3}x + 8 \] The slope \(m\) is \(-\frac{1}{3}\). 2. Since parallel lines have the same slope, the slope of the new line is also \(-\frac{1}{3}\). 3. Use the point-slope form of the equation to find the new line’s equation: \[ y - y_1 = m(x - x_1) \] Substitute \(m = -\frac{1}{3}\), \(x_1 = -8\), and \(y_1 = -8\). \[ y + 8 = -\frac{1}{3}(x + 8) \] 4. Simplify to get the equation in slope-intercept form: \[ y + 8 = -\frac{1}{3}x - \frac{8}{3} \] \[ y = -\frac{1}{3}x - \frac{8}{3} - 8 \] \[ y = -\frac{1}{3}x - \frac{8}{3} - \frac{24}{3} \] \[ y = -\frac{1}{3}x - \frac{32}{3} \] So, the equation in slope-intercept form is: \[ y = -\frac{1}{3}x - \frac{32}{3} \] 5. Convert this equation into standard form \(Ax + By = C\). Multiply through by 3 to eliminate fractions: \[ 3y = -x - 32 \] Rearrange to get: \[ x + 3y = -32 \] So, the equation in standard form is: \[ x + 3y = -32 \]