Problem 10 - Precalculus 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 equation of the line passing through (-1,3) and parallel to the line \(3x - 2y - 5 = 0\) is \(3x - 2y + 9 = 0\).

Step 1: Find the slope of the given line

Rearrange the equation \(3x - 2y - 5 = 0\) to the form \(y=mx+b\), where \(m\) is the slope. That gives us \(2y=3x+5\), then \(y=\frac{3}{2}x+\frac{5}{2}\). So the slope \(m\) of the line is \(\frac{3}{2}\).

Step 2: Write the equation for the unknown line in point-slope form

The point-slope form of the equation of a line is \(y - y_1 = m(x - x_1)\) where \((x_1, y_1)\) is a point on the line and \(m\) is the slope. The unknown line passes through the point (-1,3) and has the same slope as the given line which is \(\frac{3}{2}\). Thus substituting the given values into the equation gives us: \(y - 3 = \frac{3}{2}(x -(-1))\), which simplifies to \(y - 3 = \frac{3}{2}x + \frac{3}{2}\).

Step 3: Convert the equation into general form

The general form of the equation of a line is \(Ax + By + C = 0\). We can convert our equation into this form by rearranging and multiplying through by 2 to eliminate fractions: \(2(y - 3) = 3x + 3\) which gives \(2y - 6 = 3x + 3\). Finally, we rearrange to \(3x - 2y + 9 = 0\).

Key Concepts

Line Equation

A line equation is a mathematical expression that describes a straight line on a coordinate plane. It relates the x and y coordinates of every point on the line. One popular form of a line equation is the **point-slope form**: \[ y - y_1 = m(x - x_1) \] Here, \((x_1, y_1)\) is a specific point on the line and \(m\) represents the slope, which is a measure of the line's steepness. This form is especially useful when you know:

• A point on the line
• The slope of the line

In our original exercise, the point-slope form helps us write the equation of a line passing through the given point, \((-1, 3)\), with a slope of \(\frac{3}{2}\). The result is a clear mathematical description of where the line is on the graph.

Parallel Lines

Parallel lines are lines in a plane that never meet. They have the exact same slope. If two lines are parallel, they ascend or descend at the same rate and will never intersect. In the exercise, we need to create a line that is parallel to another given by: \[3x - 2y - 5 = 0\] We first need to find the slope of this line by rearranging its equation to the slope-intercept form \(y = mx + b\), where \(m\) stands for the slope. This gives us a slope \(m = \frac{3}{2}\). Using this, and knowing that our new line passes through \((-1, 3)\), ensures these lines are parallel. Knowing about parallel lines and their properties makes it easier to solve problems involving them. Always look for identical slopes for paraphrases.

General Form of a Line

The general form of a line equation, \(Ax + By + C = 0\), is another common way to express lines, often used for algebraic manipulation. Each term represents:

• \(A\) and \(B\): coefficients of \(x\) and \(y\)
• \(C\): constant term

This form presents a nice symmetrical pattern and can handle vertical lines, something other forms struggle with. Transitioning from point-slope to general form involves simplifying, rearranging, and sometimes eliminating fractions. In the exercise, we convert from point-slope form \(y - 3 = \frac{3}{2}(x + 1)\) to general form \(3x - 2y + 9 = 0\) through algebraic manipulation. Understanding the general form is crucial for recognizing and solving complex line equations and showing geometric relationships.