Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Write each equation in the form \(y=m x+b .\) (A suggested window for a comprehensive graph of the equation is given. $$\begin{aligned} &-0.23 x-0.46 y=0.82\\\ &[-5,5] \text { by }[-5,5] \end{aligned}$$$$\begin{aligned} &1.2 x+1.6 y=5.0\\\ &[-6,6] \text { by }[-4,4] \end{aligned}$$

Step-by-Step Solution

Answer

Equation 1: \(y = \frac{1}{2}x - 1.783\). Equation 2: \(y = -0.75x + 3.125\).

Step 1: Rearrange equation 1 into slope-intercept form

Start with the given equation, \(-0.23x - 0.46y = 0.82\).To solve for \(y\), first isolate \(y\) by moving the \(-0.23x\) term to the other side:\(-0.46y = 0.23x + 0.82\).Then, divide all terms by \(-0.46\) to solve for \(y\):\(y = \frac{-0.23}{-0.46}x + \frac{0.82}{-0.46}\).Simplify the fraction to get:\(y = \frac{1}{2}x - 1.783\). This is in the form \(y = mx + b\).

Step 2: Rearrange equation 2 into slope-intercept form

Start with the second equation,\(1.2x + 1.6y = 5.0\).To solve for \(y\), first, move the \(1.2x\) term to the other side:\(1.6y = -1.2x + 5.0\).Next, divide all terms by \(1.6\) to isolate \(y\):\(y = \frac{-1.2}{1.6}x + \frac{5.0}{1.6}\).Simplify the fractions to get:\(y = -0.75x + 3.125\). This equation is now in the form \(y = mx + b\).

Key Concepts

Linear Equations

Linear equations are mathematical expressions that form a straight line when graphed on a coordinate plane. They often come in the format of \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. These equations show a direct relationship between the two variables, \(x\) and \(y\). A linear equation can be rewritten into various forms, with the slope-intercept form being one of the most common.This form, represented as \(y = mx + b\), clearly illustrates the slope \(m\), which describes the angle or steepness of the line, and the y-intercept \(b\), which is where the line crosses the y-axis. Grasping this concept helps with understanding how one variable affects another in linear relationships, providing foundational knowledge necessary for solving more complex algebraic problems.

Algebraic Manipulation

Algebraic manipulation involves rearranging algebraic expressions to solve for a particular variable or to simplify the equation. It is a skill that enables you to isolate a variable on one side of the equation. In the given exercises, we converted equations from a standard form of \(ax + by = c\) to the slope-intercept form \(y = mx + b\).To do this:

• Move terms involving the variable to be isolated by adding or subtracting them across the equation.
• Simplify fractions by dividing both sides by the coefficient of the variable you want to isolate.
• Keep equations balanced by performing the same operation to both sides.

Mastering this procedure is pivotal because it forms the basis for solving real-world problems where relationships between variables need to be understood or predicted.

Graphing Equations

Graphing equations helps visualize the relationship between variables. A linear equation, when graphed, shows a straight line on the Cartesian plane. This is particularly effective in the slope-intercept form \(y = mx + b\), where graphing becomes straightforward.Here's how you can graph these equations:

• Start by identifying the y-intercept \(b\) and plotting it on the y-axis.
• Use the slope \(m\), which represents the rise over run, to determine the inclination of the line. For example, a slope of \(-0.75\) indicates that for every unit you move right along the x-axis, you move down \(0.75\) units along the y-axis.
• Draw a straight line through these points to complete the graph.

Understanding graphing helps to visually interpret solutions and comprehend how changes in variables influence outcomes.