Problem 26 - Introductory Algebra for College Students

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises. Then use the point-slope form of the equation to write the slope-intercept form of the equation. Passing through \((1,-3)\) with \(x\) -intercept \(=-1\)

Step-by-Step Solution

Answer

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

Step 1: Finding the slope

The slope of the line \(m\) can be found using the point \((x_1, y_1) = (1, -3)\) and the x-intercept \((x_2, y_2) = (-1, 0)\). Use the formula for calculating the slope: \(m = \frac{y_2 - y_1}{x_2 - x_1}\). The result is: \(m = \frac{0 - (-3)}{-1 - 1} = -1.5\)

Step 2: Writing Point-Slope form of the line

The point-slope form of the line is given by the formula: \(y - y_1 = m(x - x_1)\). Substituting the values for \(x_1\), \(y_1\) and \(m\) from step 1, the point-slope form of the line is \((y - (-3)) = -1.5(x - 1)\) or \(y + 3 = -1.5x + 1.5\).

Step 3: Writing Slope-Intercept form of the line

The slope-intercept form is \(y = mx + b\). Rearranging the equation obtained in step 2 to slope-intercept form involves isolating \(y\) on one side of the equation. Doing this results in: \(y = -1.5x + 1.5 - 3\) or \(y = -1.5x - 1.5\).

Key Concepts

Slope-Intercept Form

The slope-intercept form is one of the easiest methods to write and read a linear equation. The formula looks like this: \( y = mx + b \), where

• \( m \) represents the slope of the line, showing how steep the line is,
• \( b \) is the y-intercept, revealing where the line crosses the y-axis.

Understanding slope-intercept form is crucial because it provides a straightforward approach to graphing lines on a coordinate plane. You can identify the start point on the y-axis through \( b \) and then use the slope \( m \) to find other points. If you're given a linear equation in a different format, it's quite useful to rewrite it in slope-intercept form to quickly grasp the visual representation of the line.

Slope of a Line

The slope of a line is a measure of its steepness and direction. Mathematically, it's defined as the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. Expressed as an equation, this is \( m = \frac{\Delta y}{\Delta x} \) or \( m = \frac{y_2 - y_1}{x_2 - x_1} \), where \( (x_1, y_1) \) and \( (x_2, y_2) \) are coordinates of two distinct points on the line.

A positive slope means that the line rises, whereas a negative slope indicates that the line falls as you move from left to right. A slope of zero corresponds to a horizontal line, and when the slope is undefined (due to a division by zero), the line is vertical. The concept of slope is foundational in algebra for understanding and interpreting linear relationships.

Algebraic Equations

An algebraic equation is like a scale where two expressions balance each other out. Each side of the equal sign has an expression, and the idea is to find the value(s) of the unknowns that make the equation true. In the world of algebra, these unknowns are usually depicted by letters, like 'x' or 'y', and these letters can represent numbers.

There are different types of algebraic equations, with varying levels of complexity. Linear equations, which include the point-slope and slope-intercept forms, are an example where the highest power of the unknown is one (e.g., \( ax + b = 0 \)). To solve these equations, you often perform operations such as adding, subtracting, multiplying, dividing or factoring to isolate the unknown and find its value.

Linear Equations

The linear equation is a statement of equality that involves only linear functions of one or more variables. The general form of a linear equation in two variables (x and y) is given by \( Ax + By + C = 0 \), with A, B, and C representing constants. The key characteristic of linear equations is their graph on the coordinate plane, which is always a straight line.

In practical applications, linear equations can describe an endless variety of situations, from predicting profits in a business to understanding rates of change in science. They're pivotal when dealing with proportionate relationships, and converting between different forms of linear equations (like point-slope and slope-intercept) can help in various algebraic procedures, such as graphing a line or solving systems of equations.