Problem 10 - College Algebra Essentials

Question

Use the given conditions to write an equation for each line in point-slope form and general form. Passing through \((-1,3)\) and parallel to the line whose equation is \(3 x-2 y-5=0\)

Step-by-Step Solution

Answer

The point-slope form of the line is \(y - 3 = 1.5(x + 1)\) and the general form is \(-1.5x + y - 4.5 = 0\).

Step 1: Extract Slope From Given Line

The equation of the give line is in the standard form Ax + By + C = 0. Convert it into slope-intercept form to find the slope. The slope-intercept form is y = mx + c where m is the slope. Thus we can re-arrange the given equation as follows: \(2y = 3x - 5\) or \(y = 1.5x - 2.5\). The slope of this line is 1.5.

Step 2: Understand the Concept of Parallel Lines

Parallel lines have the same slope. Therefore, the new line that is parallel to the given line will also have a slope of 1.5.

Step 3: Incorporate Point in the Equation

Use the point-slope form of line equation which is (y - y1) = m(x - x1) where (x1, y1) is the point the line passes through and m is the slope. Our line therefore is \(y - 3 = 1.5(x - (-1))\), simplifying to \(y - 3 = 1.5x + 1.5\).

Step 4: Simplify Equation to General Form

Subtract 1.5x from both sides to get it into the general form Ax + By + C = 0. The equation, when simplified, becomes \(-1.5x + y - 4.5 = 0\).