Problem 32 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and slope-intercept form. Passing through \((-3,6)\) and \((3,-2)\)

Step-by-Step Solution

Answer

The equation of the line passing through the points \((-3,6)\) and \((3,-2)\) in point-slope form is \(y - 6 = -4/3(x + 3)\), and in slope-intercept form it is \(y = -4/3x + 2\).

Step 1: Determining the slope

Identify the coordinates of the two points, \((-3,6)\) and \((3,-2)\). The slope (m) of the line passing through these points can be found by the formula: m = \((y2 - y1) / (x2 - x1)\) where \((x1, y1)\) are the coordinates of the first point and \((x2, y2)\) are the coordinates of the second point. Plugging in the given points provides: m = \((-2 - 6) / (3 - (-3)) = -8 / 6 = -4/3\).

Step 2: Writing the point-slope form

The point-slope form of a line is given by: \(y - y1 = m(x - x1)\), where m is the slope and \((x1, y1)\) is a point on the line. Plug the slope and the coordinates of either of the two points into this formula. Let's use the point \((-3,6)\) for example: \(y - 6 = -4/3(x - (-3))\). This simplifies to: \(y - 6 = -4/3(x + 3)\).

Step 3: Transform to the slope-intercept form

The slope-intercept form is given by \(y = mx + b\). Transform the point-slope form from step 2 to the slope-intercept form by distributing -4/3 through the parentheses and isolating y on one side of the equation: \(y = -4/3x - 4 + 6\), which simplifies to: \(y = -4/3x + 2\).