Problem 12 - Introductory and Intermediate Algebra for College Students 4th

Question

Write the point-slope form of the equation of the line satisfying each of the conditions in Exercises \(1-28 .\) Then use the point-slope form of the equation to write the slope-intercept form of the equation. Slope \(=\frac{1}{3},\) passing through the origin

Step-by-Step Solution

Answer

The point-slope form of the line is \(y=\frac{1}{3}*x\) and the slope-intercept form of the equation is \(y=\frac{1}{3}x\).

Step 1: Point-Slope Form Equation

For slope=m=\(\frac{1}{3}\) and the point through which it passes being the origin (0,0), the point-slope form equation will be: \( y-y1=m(x-x1) => y-0=\frac{1}{3}(x-0) => y=\frac{1}{3}*x\).

Step 2: Slope-Intercept Form Equation

From the point-slope form equation in Step 1, the equation can be written in slope-intercept form (y=mx+b) as it already matches that format. So the slope-intercept form of the equation is \(y=\frac{1}{3}x\).

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is one of the most common ways to express the equation of a line. It is given as \(y = mx + b\), where:

• \(y\) is the dependent variable or the output of the function.
• \(m\) represents the slope of the line.
• \(x\) is the independent variable or the input of the function.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

Understanding and identifying each of these components are key in graphing and interpreting linear equations.
The slope \(m\) can tell you how steep the line is. A positive slope means the line ascends from left to right, whereas a negative slope means it descends. The y-intercept \(b\) is the starting point of the line when \(x = 0\).
This format is popular in algebra because it provides a straightforward way to graph lines on a coordinate plane. It allows you to easily determine how the line behaves just by looking at its equation.

Linear Equations

Linear equations are mathematical expressions that describe a straight line on a graph. They are fundamental in algebra and are commonly written in the form \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants.
Linear equations can be rewritten into different forms, such as slope-intercept form or point-slope form, depending on what information is known:

• The slope-intercept form \(y = mx + b\) highlights the slope and y-intercept of the line.
• The point-slope form \(y - y_1 = m(x - x_1)\) is useful when you know the slope and a specific point \((x_1, y_1)\) the line passes through.

Linear equations are essential because they model real-world situations where there is a constant rate of change. For instance, they can represent the relationship between distance and time when speed is constant.
By understanding linear equations, students can solve problems involving trends, patterns, and predictions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They are used extensively to represent mathematical relationships in algebra. A simple form of an algebraic expression is \(3x + 2\) where:

• \(3\) is a coefficient multiplying the variable \(x\).
• \(x\) is the variable representing an unknown value.
• \(2\) is a constant term.

These expressions serve as the building blocks for creating equations and inequalities. They allow you to describe rules and patterns within mathematics concisely.
Understanding algebraic expressions is crucial, as they facilitate solving equations and graphing functions. By simplifying or manipulating these expressions, students can uncover solutions to complex problems and identify relationships between different quantities.