Problem 34 - Algebra Form and Function

Question

Solve for the indicated variable. Assume all constants are non-zero. $$ a b=c, \text { for } b $$

Step-by-Step Solution

Answer

Question: Given the equation \(ab=c\), solve for b in terms of a and c. Answer: \(b=\frac{c}{a}\)

Step 1: Identify the target variable

The target variable we want to solve for is b, and we're given the equation: $$ a b=c $$

Step 2: Isolate the target variable

To isolate b, we want to eliminate any terms involving a from the left side of the equation. Since a is multiplied by b, we can simply divide both sides by a to achieve isolation of b: $$ \frac{a b}{a}=\frac{c}{a} $$

Step 3: Simplify the equation

On the left side of the equation, we have the term \(ab/a\) which, when simplified is \(b/a\). Hence the simplified equation is: $$ b=\frac{c}{a} $$

Step 4: Write the final answer

Solving for b in the equation \(ab=c\), we find that: $$ b=\frac{c}{a} $$This is the final answer, and b is now expressed in terms of a and c.

Key Concepts

Isolation of Variables

When solving equations, one of the main goals is to isolate the variable we are interested in. In this case, we want to solve for \(b\) in the equation \(ab = c\).
Isolation means getting the variable by itself on one side of the equation. For example, if \(b\) is the variable we need to find, we want \(b\) to be alone on one side. This means removing any constants or coefficients, which might be linked with \(b\) by operations like multiplication or addition.
There are some steps to achieve this:

• Identify the variable: First, pinpoint which variable you want to solve for.
• Undo operations: Perform operations, like division, to remove attached constants or coefficients.
• Maintain balance: To keep the equation balanced, whatever you do to one side must be done to the other.

This approach is crucial in algebra, providing the foundation to solve and simplify various equations effectively.

Basic Algebra

Basic algebra involves understanding and manipulating variables and constants to solve equations. At the core of algebra is the ability to perform operations such as addition, subtraction, multiplication, and division to rearrange equations. These operations are guided by certain rules and properties.
For example, the equation \(ab = c\) requires you to solve for one variable in terms of others. The operations you will use must respect the equation's balance. Specifically, you can multiply or add the same thing on both sides to maintain equality.
Key concepts to remember in basic algebra include:

• Commutative Property: The order of addition or multiplication doesn't change the result, e.g., \(a + b = b + a\).
• Associative Property: The grouping of numbers doesn't matter for addition or multiplication, e.g., \((a + b) + c = a + (b + c)\).
• Distributive Property: Multiplication can be distributed over addition, e.g., \(a(b + c) = ab + ac\).

Utilizing these principles helps in easily managing and isolating variables in algebraic equations.

Division in Equations

Division is a fundamental operation used to solve equations, especially when you need to isolate a variable that is multiplied by another term. In the equation \(ab = c\), division is used to separate the variable \(b\) from \(a\).
To do this, you divide both sides of the equation by \(a\), resulting in \(b = \frac{c}{a}\). This division cancels out \(a\) on the left side, leaving \(b\) by itself. It's important to ensure that \(a\) is non-zero to avoid division by zero, which is undefined.
Here are some tips when using division in solving equations:

• Equal Treatment: Always divide every term in the equation equally; if you divide the left side by \(a\), do the same to the right side.
• Watch for Division by Zero: Ensure that the divisor is not zero as it will lead to an undefined result.
• Simplify: After dividing, simplify the equation if necessary to clearly express the variable.

Mastering division in equations leads to a more straightforward process of finding solutions in algebra, helping to clearly and effectively isolate and solve for the desired variable.