Problem 19 - Algebra for College Students

Question

Solve each formula for the specified variable.\(a x+b=0\)(linear equation in \(x\) ) (a) for \(x\) (b) for \(a\)

Step-by-Step Solution

Answer

For part (a): \(x = \frac{-b}{a}\). For part (b): \(a = \frac{-b}{x}\).

Step 1: Identify and isolate the variable

First, we need to isolate the variable we want to solve for. In part (a), we need to solve for \(x\). Start by subtracting \(b\) from both sides of the equation \(a x + b = 0\). This gives us \(a x + b - b = 0 - b \) which simplifies to \( a x = -b \).

Step 2: Solve for \(x\)

Next, divide both sides of the equation by \(a\) to isolate \(x\). This gives us \( x = \frac{-b}{a} \).

Step 3: Confirm the solution for \(x\)

The solution to the equation \(a x + b = 0\) for \(x\) is \( x = \frac{-b}{a} \).

Step 4: Solve for \(a\)

Now in part (b), we need to solve for \(a\). Start with the given equation \(a x + b = 0\). Subtract \(b\) from both sides: \(a x = -b\).

Step 5: Isolate and solve for \(a\)

To isolate \(a\), divide both sides by \(x\). This gives us \( a = \frac{-b}{x} \).

Step 6: Confirm the solution for \(a\)

The solution to the equation \(a x + b = 0\) for \(a\) is \( a = \frac{-b}{x} \).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is a fundamental skill in solving equations. It involves rearranging and simplifying equations to isolate a variable. Let's explore how this works by solving the linear equation given in the exercise:

Start with the equation: \( ax + b = 0 \)
Our goal is to solve for the variables specified. To solve for \( x \) in part (a), we use algebraic manipulation to isolate \( x \):

Subtract \( b \) from both sides: \[ ax + b - b = 0 - b \] This simplifies to: \[ ax = -b \]
Now, we divide both sides by \( a \): \[ x = \frac{-b}{a} \]
Using these steps, we can isolate \( x \). Algebraic manipulation ensures that each side of the equation remains equal while we change its form to make solving easier. Following the same principles, we solve for \( a \) in part (b).

Begin with the same equation: \( ax + b = 0 \)
Again, subtract \( b \): \[ ax = -b \]
Next, divide both sides by \( x \): \[ a = \frac{-b}{x} \]
This systematic method of rearranging equations showcases the power of algebraic manipulation in solving for different variables.

Understanding Formulas

Formulas are mathematical expressions that show the relationship between different variables. In the given exercise, the formula is \( ax + b = 0 \). Here, \( a \), \( x \), and \( b \) are variables that can represent various values.

When using formulas:

• Identify all variables within the formula.
• Understand how these variables interact.
• Use algebraic manipulation to solve for the desired variable.

Let's break down the formula further.

For (a), we needed to find what \( x \) equals:
Start with: \( ax + b = 0 \)
Subtract \( b \): \[ ax = -b \]
Divide by \( a \): \[ x = \frac{-b}{a} \]

For (b), we solved for \( a \):
Start with: \( ax + b = 0 \)
Subtract \( b \): \[ ax = -b \]
Divide by \( x \): \[ a = \frac{-b}{x} \]

This shows the versatility of formulas. By working through these steps, we see how changes in one part of the formula affect the others.

Variable Isolation

Variable isolation is a key technique in solving equations. It means getting the variable of interest alone on one side of the equation.

To isolate a variable:

• Perform the same operation on both sides of the equation.
• Simplify the equation step-by-step.
• Check each step to maintain equality.

In part (a) of the exercise, our goal was to isolate \( x \):
Start with: \( ax + b = 0 \)
Subtract \( b \): \[ ax = -b \]
Divide by \( a \): \[ x = \frac{-b}{a} \]

In part (b), we isolated \( a \):
Start with: \( ax + b = 0 \)
Subtract \( b \): \[ ax = -b \]
Divide by \( x \): \[ a = \frac{-b}{x} \]

By isolating the variable, we make it the subject of the formula. This technique is crucial for solving equations in algebra and beyond.