Problem 61 - College Algebra

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-8,-10)\) and parallel to the line whose equation is \(y=-4 x+3\)

Step-by-Step Solution

Answer

The equation of the line passing through (-8,-10) and parallel to the given line is \(y = -4x + 22\) in slope-intercept form.

Step 1: Identifying the Slope of the Original Line

Parallel lines share the same slope. In the equation y = -4x + 3, the coefficient of x is the slope. Therefore, the slope of the line parallel to the given line, is -4.

Step 2: Use the Point-Slope Form of a Line

The point-slope form of a line is \(y - y_1 = m(x - x_1)\), where m is the slope and \((x_1, y_1)\) is a point on the line. Using the calculated slope -4 and given point (-8,-10), the points are substituted into equation yielding \(y - (-10) = -4(x - (-8))\).

Step 3: Simplify the Equation into Slope-Intercept Form

The equation can be simplified to its slope-intercept form, y = mx + b, by distributing -4 into \(x - (-8))\), and simplifying \(y - (-10)\) to \(y + 10)\). This yields \(y + 10 = -4x + 32\). By then solving for y, the final equation in slope-intercept form is \(y = -4x + 22\).