Problem 13 - College Algebra with Corequisite Support

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, the equation can be written as a linear function: \(y = -\frac{1}{10}x + \frac{3}{10}\).

Step 1: Identify the form of the equation

The given equation is \(-\frac{x-3}{5}=2y\). This needs to be rearranged to see if it can be written in the linear form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.

Step 2: Isolate y in the equation

Begin by multiplying both sides of \(-\frac{x-3}{5}=2y\) by 5 to clear the fraction.\[ -1(x-3) = 10y \]Now distribute the \(-1\) to get:\[ -x + 3 = 10y \]Finally, divide by 10 to solve for \(y\):\[ y = -\frac{1}{10}x + \frac{3}{10} \]

Step 3: Determine if the equation is a linear function

The rearranged equation \(y = -\frac{1}{10}x + \frac{3}{10}\) is in the standard linear form \(y = mx + b\). Here, the slope \(m\) is \(-\frac{1}{10}\) and the y-intercept \(b\) is \(\frac{3}{10}\). Since it is written as \(y = mx + b\), it is a linear function.

Key Concepts

Equation Rearrangement

Rearranging an equation is a critical step in math that simplifies how we view functions. By changing an equation's format, we can gain better insights into what the equation represents and how it behaves. In the exercise, we started with the equation -\(\frac{x-3}{5}=2y\), which initially appears complex. To simplify it, we rearrange the equation into a more user-friendly format.

Rearranging involves several steps:

• Clearing fractions or denominators by multiplying both sides by the same value, making the equation easier to handle.
• Distributing values if necessary, helping to eliminate parenthesis and combining like terms.
• Isolating the variable of interest (in our case, \(y\)), putting the equation into a recognizable form.

In this scenario, by solving for \(y\), we turned the equation into \(y = -\frac{1}{10}x + \frac{3}{10}\). This makes it easier to identify the characteristics of the function, such as its slope and intercept.

Slope-Intercept Form

The slope-intercept form of a line is a key concept in understanding linear functions. It is represented as \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. This form gives a clear depiction of how the line will behave on a graph.

Understanding slope and intercept:

• Slope \(m\): Represents the rate at which the line rises or falls. A positive slope means the line ascends, while a negative slope means it descends.
• Y-intercept \(b\): The point where the line crosses the y-axis. It tells you the starting point of the line when \(x\) is zero.

In the rearranged equation \(y = -\frac{1}{10}x + \frac{3}{10}\), the slope \(-\frac{1}{10}\) tells us that for each unit increase in \(x\), \(y\) decreases by \(\frac{1}{10}\). The y-intercept \(\frac{3}{10}\) indicates the line crosses the y-axis at the point \(0, \frac{3}{10}\).

This representation helps students easily draw and understand the line's direction and position on the graph.

Solving Equations

Solving equations is the process of finding the values of variables that satisfy the equation. It involves manipulating the equation to isolate the variable, making it possible to find exact numerical values.

Basic steps involved in solving equations include:

• Cancelling out terms: Use addition or subtraction to eliminate unwanted terms from one side of the equation.
• Dividing or multiplying: Apply these operations to both sides of the equation to simplify and isolate the variable.
• Substitution (if necessary): Replace variables or terms if the equation involves more complex expressions.

In our exercise, we multiplied by 5 to eliminate the fraction, distributed the -1, and then divided by 10 to solve for \(y\). Solving the equation in this structured manner ensures that each step builds on the previous one, maintaining the equation's integrity and leading to the final result \(y = -\frac{1}{10}x + \frac{3}{10}\). These methods ensure that you reach the correct solution efficiently and accurately.