Problem 43 - A Graphical Approach to College Algebra

Question

Find the equation of the line satisfying the given conditions, giving it in slope-intercept form if possible. Through \((-1,4),\) parallel to \(x+3 y=5\)

Step-by-Step Solution

Answer

The equation is \(y = -\frac{1}{3}x + \frac{11}{3}\).

Step 1: Identify the slope from the given equation

The given line is represented by the equation \(x + 3y = 5\). To find the slope, rewrite it in slope-intercept form, \(y = mx + b\). Start by isolating \(y\): \(3y = -x + 5\), then \(y = -\frac{1}{3}x + \frac{5}{3}\). The slope \(m\) of this line is \(-\frac{1}{3}\).

Step 2: Determine the slope of the parallel line

Lines that are parallel have the same slope. Since our new line needs to be parallel to the line from Step 1, it will also have a slope of \(-\frac{1}{3}\).

Step 3: Use the point-slope form to find the equation

Insert the point \((-1, 4)\) and the slope \(-\frac{1}{3}\) into the point-slope form equation of a line: \( y - y_1 = m(x - x_1) \). Here, \(y_1 = 4\), \(m = -\frac{1}{3}\), and \(x_1 = -1\). So the equation becomes: \( y - 4 = -\frac{1}{3}(x + 1)\).

Step 4: Simplify to slope-intercept form

Begin by distributing the slope on the right side: \( y - 4 = -\frac{1}{3}x - \frac{1}{3}\). Then solve for \(y\) to put the equation in slope-intercept form: \( y = -\frac{1}{3}x - \frac{1}{3} + 4\). Convert \(4\) to thirds: \(4 = \frac{12}{3}\). Add: \(-\frac{1}{3} + \frac{12}{3} = \frac{11}{3}\). So the final equation is \(y = -\frac{1}{3}x + \frac{11}{3}\).

Key Concepts

Equation of a Line

An equation of a line is a mathematical statement that describes all possible points forming a straight line on a plane. There are several ways to express the equation of a line, but one of the most popular forms is the slope-intercept form.

- The slope-intercept form of a line's equation is written as \(y = mx + b\), where: - \(m\) represents the slope, which describes the steepness of the line. - \(b\) is the y-intercept, the point where the line crosses the y-axis. To find the equation of a line, you'll need to know either:

- Two points on the line - One point on the line and the slope.

Understanding these components helps in sketching the line and analyzing its behavior. This form is particularly useful for graphing because it readily shows the line's slope and intercept.

Parallel Lines

Parallel lines are lines in a plane that never meet. They have the same slope but different intercepts. Knowing the slope of one line helps you find a parallel line by ensuring their slopes match.

- If line A has an equation \(y = mx + b\), any line parallel to line A will have an equation in the form \(y = mx + c\), where \(b\) is not equal to \(c\).In geometry, the concept of parallel lines is essential because it helps in forming grids and understanding spatial relationships. In algebra, this concept guarantees that lines with equal slopes never intersect, thereby verifying their parallelism.

Point-Slope Form

The point-slope form is another way to express the equation of a line. It is especially useful when you have a point on the line and the slope available.

- The general expression for point-slope form is given by \(y - y_1 = m(x - x_1)\), where: - \((x_1, y_1)\) is a point on the line. - \(m\) is the slope of the line.This form is handy for quickly forming an equation without first converting points or needing additional computation. It can then be converted to a slope-intercept form by simplifying the terms, offering versatility in problem-solving.