Problem 80 - Intermediate Algebra

Question

Solve for the specified variable. $$ A x+B y=C \quad \text { for } x $$

Step-by-Step Solution

Answer

The solution for \(x\) is \(x = \frac{C - B y}{A}\).

Step 1: Isolate the term with x

To solve for \(x\), we first need to isolate the term containing \(x\). Start by subtracting \(B y\) from both sides of the equation. This gives us:\[A x = C - B y\]

Step 2: Solve for x

Now we need to solve for \(x\) by isolating it completely. Since \(A x\) means \(A\) times \(x\), we can divide both sides of the equation by \(A\) to get:\[x = \frac{C - B y}{A}\]

Key Concepts

Solving for a Variable

In algebra, solving for a variable means finding the value of that variable that makes an equation true. An equation can have one or more variables, and when solving for one specific variable, you are adjusting the equation to express this variable in terms of the others.
For example, given an equation like \( A x + B y = C \) and being asked to solve for \( x \), you're essentially manipulating the equation so that \( x \) stands alone on one side. This often involves performing operations such as addition, subtraction, multiplication, or division to isolate the variable. This process helps in understanding how one variable depends on others.

• Start by identifying the term you need to isolate.
• Perform operations to get this variable by itself on one side of the equation.
• Ensure the operations you perform are all valid, maintaining the equality of the equation throughout the process.

Linear Equations

A linear equation is an equation in which each term is either a constant or the product of a constant and a single variable. In their simplest form, these equations take on the shape of a straight line when graphed on a coordinate plane. For our exercise, the equation \( A x + B y = C \) represents a two-variable linear equation of the form \( ext{Variable 1} + ext{Variable 2} = ext{Constant} \). These equations are easier to manipulate because the highest power of the variables is 1, meaning there's no squaring, cubing, or higher-level exponents involved.

• Linear equations help in predicting outputs based on inputs due to their predictable nature.
• They are often used as an introduction to more complex algebraic equations due to their simplicity.
• Understanding linear equations is foundational for advanced math concepts.

Isolation of Variables

The process of isolating a variable is essential to solve equations efficiently. It involves rearranging the equation so that the variable of interest is on one side of the equal sign, completely by itself. To isolate \( x \) in our given equation \( A x + B y = C \), you start by removing the terms that are not associated with \( x \). Initially, you subtract \( B y \) from both sides, which gives you \( A x = C - B y \). Then, you divide both sides by \( A \) to finally get \( x = \frac{C - B y}{A} \). This step-by-step method ensures clarity and helps prevent mistakes.

• Begin by moving other terms away from the variable you want to isolate.
• Use inverse operations like subtraction or division to further separate the variable.
• Check your work by substituting the solution back into the original equation to verify correctness.