Problem 13 - College Algebra

Question

For the following exercises, determine whether the equation of the curve can be written as a linear function. $$ -\frac{x-3}{5}=2 y $$

Step-by-Step Solution

Answer

Yes, it can be written as a linear function: \(y = -\frac{x}{10} + \frac{3}{10}\).

Step 1: Rearrange the Original Equation

The given equation is \[-\frac{x-3}{5} = 2y\] First, we want to isolate \(y\) on one side of the equation. Multiply both sides by 5 to eliminate the denominator:\[-(x-3) = 10y\]

Step 2: Distribute and Simplify

Next, distribute the minus sign on the left side of the equation:\[-x + 3 = 10y\] Now, solve for \(y\) by dividing each term by 10 to make the equation linear in terms of \(y\):\[y = -\frac{x}{10} + \frac{3}{10}\] Thus, the equation simplifies to this linear form.

Step 3: Identify the Linear Function

The equation \[y = -\frac{x}{10} + \frac{3}{10}\] is in the form \(y = mx + b\), where \(m = -\frac{1}{10}\) and \(b = \frac{3}{10}\). This confirms that the equation represents a linear function, with \(m\) as the slope and \(b\) as the y-intercept.

Key Concepts

Understanding Slope and Y-Intercept

A linear function is typically expressed in the form of the equation \( y = mx + b \). This format is called the slope-intercept form, where \( m \) represents the slope and \( b \) is the y-intercept. Here's what these components mean:

• Slope \( (m) \): This tells us how steep the line is. A positive slope means the line goes upward as you move from left to right, whereas a negative slope means it descends.
• Y-Intercept \( (b) \): This is the point where the line crosses the y-axis. Essentially, it shows the value of \( y \) when \( x = 0 \).

In the equation \( y = -\frac{x}{10} + \frac{3}{10} \), the slope \( m \) is \( -\frac{1}{10} \), indicating a line that gently descends from left to right. The y-intercept \( b \) is \( \frac{3}{10} \), showing that the line crosses the y-axis at \( 0.3 \). Understanding these components helps us visualize the line even before we graph it.

Mastering Equation Manipulation

Manipulating equations is all about altering the form of the equation to make it easier to work with or understand. We often do this in steps, each involving basic algebraic operations like addition, subtraction, multiplication, or division.For example, to simplify \(-\frac{x-3}{5} = 2y\), the first step is to remove any fractions. We do this by multiplying all terms by 5, eliminating the denominator:\[ -(x-3) = 10y \]Next, to simplify further, we distribute the negative sign across the parentheses, making the equation clearer:\[ -x + 3 = 10y \]Each of these operations keeps the equation balanced as long as performed on both sides. This type of manipulation is crucial for turning a complicated equation into something usable or recognizable, such as turning it into its slope-intercept form.

Solving Algebraic Equations

Solving algebraic equations involves isolating the variable you're solving for—in many cases, \( y \), to express it in terms of other variables or constants. This is a step-by-step process that often requires rearranging terms and applying various operations.In our original problem, the aim was to express \( y \) as a function of \( x \). Here's how it was done:1. After the initial equation was adjusted to \(-x + 3 = 10y\), the next step was dividing every term by 10: \[ y = -\frac{x}{10} + \frac{3}{10} \]2. This operation simplifies or 'solves' the equation for \( y \) and results in a clear linear form.This systematic methodical process underscores the importance of maintaining equation balance. By isolating \( y \), we connect the equation to practical applications, showing how different values of \( x \) affect \( y \)."}]}]}​​