Problem 128 - College Algebra with Modeling and Visualization

Question

Describe verbally how to solve ax + b = 0. What assumptions have you made about the value of a?

Step-by-Step Solution

Answer

The equation is solved by isolating \( x \) as \( x = \frac{-b}{a} \), assuming \( a \neq 0 \).

Step 1: Understand the Equation Form

The equation given is linear and in the form \( ax + b = 0 \), where \( a \) and \( b \) are constants. Our goal is to solve for \( x \).

Step 2: Isolate the Variable Term

To solve for \( x \), we first need to isolate the term containing \( x \). This is done by subtracting \( b \) from both sides of the equation:\[ax + b - b = 0 - b \a x = -b\]

Step 3: Solve for x by Dividing

Now, we solve for \( x \) by dividing both sides of the equation by \( a \) assuming \( a eq 0 \):\[ x = \frac{-b}{a}\] This step assumes that \( a \) is not equal to zero, as division by zero is undefined.

Step 4: State the Assumptions

The primary assumption made is that \( a eq 0 \). If \( a = 0 \), the equation \( ax + b = 0 \) simplifies to \( b = 0 \), leading either to no solution or infinitely many solutions if \( b eq 0 \) or \( b = 0 \), respectively.

Key Concepts

Equation Form

The journey to solving a linear equation begins with understanding its form. A linear equation in the form of \( ax + b = 0 \) is straightforward yet vital. Here:

• \( a \) and \( b \) represent constants. These are fixed numbers within the equation.
• \( x \) is the variable we need to solve for.

Recognizing the structure of this equation sets the foundation for finding the solution. The equation represents a straight line when graphed, hence why it's termed 'linear.' By learning this form, you'll be well-equipped to tackle any similar equation.

Isolate Variable

Isolating the variable is a crucial step when solving equations like \( ax + b = 0 \). The goal is to arrange the equation so that \( x \) stands alone on one side. Here’s the process:

• Start by eliminating \( b \) from the equation. You do this by subtracting \( b \) from both sides, transforming the equation into \( ax = -b \).
• This step shifts \( b \) away, isolating the term with \( x \). It’s essential to maintain balance by performing the same operation on both sides of the equation.

After this, you should have a simplified equation that makes it easier to solve for \( x \). Keeping \( x \) isolated is like zooming in on the variable, making the final solution step more apparent.

Constants

Constants in linear equations like \( ax + b = 0 \) are vital components that determine the equation's fixed aspects. Consider the role of both \( a \) and \( b \):

• \( a \) is crucial because it serves as a multiplier for \( x \). If \( a = 0 \), you won't have a valid linear equation for \( x \) as it eliminates the variable term entirely.
• \( b \) represents a constant shift in the equation. It adjusts where the linear relation cuts the y-axis when graphed.

A key assumption while solving is that \( a eq 0 \), ensuring that division by zero does not occur. Therefore, understanding the role of constants helps avoid pitfalls and leads to the correct solution.