Problem 25 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(1-26,\) solve each formula for the specified variable. Do you recognize the formula? If so, what does it describe? $$A x+B y=C \text { for } x$$

Step-by-Step Solution

Answer

The solution is \(x = (C - By) / A\).

Step 1: Identify the variable

The first step is to identify the variable that you are solving for, which in this case is \(x\).

Step 2: Isolate x

The goal is to isolate \(x\), so you need to remove \(By\) from the left side of the equation. Subtract \(By\) from both sides of the equation. This gives you \(Ax = C - By\).

Step 3: Solve for x

Now, divide both sides of the equation by \(A\) to solve for \(x\). The final formula for \(x\) is \(x = (C - By) / A\).

Key Concepts

Solving for a Variable

Understanding how to solve for a specific variable in an equation is crucial for success in algebra. In this exercise, the task is to solve for \(x\) in the equation \(Ax + By = C\). This means we want to express \(x\) in terms of all other variables. Let's break down the process in a simple way:

• **Identify the Target Variable**: First, determine which variable you need to solve for. Here, it is \(x\).
• **Isolate the Variable**: Begin by moving all terms that do not contain \(x\) to the other side of the equation.
• **Solve for the Variable**: Finally, perform operations to 'untie' \(x\) from other terms. This might involve addition, subtraction, multiplication, or division.

Through this step-by-step approach, solving for any variable becomes a clear and manageable process. Solving for a variable is about understanding the relationship between variables in an equation and expressing one variable as a function of others.

Equation Manipulation

Equation manipulation refers to the process of performing operations to rewrite equations. The goal is often to isolate a specific variable or simplify the equation. In the given example, to solve for \(x\), we manipulate the original equation \(Ax + By = C\) using several steps:

• **Subtract **\(By\) from both sides: This allows us to start isolating \(x\). The new equation becomes \(Ax = C - By\).
• **Divide by A**: To completely isolate \(x\), divide the entire equation by \(A\) which gives \(x = \frac{C - By}{A}\).

These manipulations maintain the balance of the equation. Remember, any operation carried out on one side must also be applied to the other side to maintain equality. This ensures the equation's original meaning is preserved, and we derive valid solutions.

Algebraic Concepts

Algebraic concepts provide the framework for understanding and solving equations like the one given. They include fundamental ideas such as variables, coefficients, and operations that bind arithmetic and algebra together:

• **Variables**: Represent unknown values. In our example, \(x\) and \(y\) are variables.
• **Coefficients**: Numbers that are multiplied by the variables. Here, \(A\) and \(B\) serve as coefficients.
• **Equations**: Mathematical statements showing that two expressions are equal, important for expressing relationships.

Mastery of these concepts allows for flexibility in handling complex equations. Recognizing how variables and coefficients interact lets you set up and solve equations efficiently. By breaking down each part of an equation, you deepen your understanding of algebra as a whole. This fundamental knowledge is essential not just for solving specific problems, but also for applying algebraic techniques in more advanced mathematics and real-world situations.