Problem 49 - Introductory and Intermediate Algebra for College Students 4th

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 parallel to \(y=-4x+3\) passing through \((-8,-10)\) is \(y = -4x - 42\).

Step 1: Find the slope of the given line

The slope is obtained from the given equation. The equation \(y=-4x+3\) is in the slope-intercept form \(y=mx+b\) where \(m\) is the slope. So, slope of the given line is -4. As parallel lines have the same slope, the slope of the line we're looking for is also -4.

Step 2: Writing the equation in point-slope form

The point-slope form of the equation of a line is \(y - y1 = m(x - x1)\) where \((x1,y1)\) is a point on the line and \(m\) is the slope of the line. Here, the point is \((-8,-10)\) and the slope is -4. Substituting these values, the point-slope form is \(y - (-10) = -4(x - (-8))\). Simplifying, we get \(y + 10 = -4(x + 8)\).

Step 3: Transforming to the slope-intercept form

To get the equation into slope-intercept form \(y=mx+b\), we just need to arrange the equation from Step 2. Distributing -4 into \((x + 8)\), the equation transforms to \(y + 10 = -4x - 32\). Finally, transfer 10 to the other side to obtain \(y = -4x - 42\).