Problem 44 - Introductory Algebra for College Students

Question

Solve each equation for \(x .\) $$y=(a-b) x$$

Step-by-Step Solution

Answer

The solution to \(y = (a - b) x\) for \(x\) is \(x = \frac{y}{a-b}\).

Step 1: Understand the initial equation

\The equation provided is \(y = (a - b) x\). The ultimate aim is to make \(x\) the subject of the equation.

Step 2: Isolate \(x\)

To achieve this, we should divide both sides of the equation by \(a - b\). This action doesn't violate any mathematical principles and allows us to solve for \(x\). This yields \(x = \frac{y}{a-b}\).

Step 3: Simplify the equation

Our solution doesn't need further simplification. Thus, the final expression for \(x\) in terms of \(y\), \(a\), and \(b\) is \(x = \frac{y}{a-b}\).

Key Concepts

Solving Equations

Solving equations is a foundational skill in math that involves finding the value of a variable that makes the equation true. In the equation \(y = (a-b) x\), the goal is to determine the value of \(x\) in terms of \(y\), \(a\), and \(b\). To begin, we need to understand that this equation is a linear equation, meaning it graphs as a straight line and has no exponents on the variable \(x\) beyond 1. We approach solving such equations systematically by applying operations that keep both sides of the equation balanced. This ensures that the equality holds true through the process.
When solving equations, it is crucial to recognize which operations are needed to isolate the variable of interest, in this case, \(x\).

Isolation Method

The isolation method is a common technique used in algebra to solve for a specific variable. The key concept here is manipulating the equation so that the variable you want to solve for is by itself on one side of the equation. In the provided equation, \(y = (a-b) x\), our goal is to isolate \(x\).
To accomplish this, we need to divide both sides of the equation by \((a-b)\).

• This division cancels out the coefficient \((a-b)\) multiplying \(x\), thus isolating it on one side of the equation.
• It is vital to ensure \((a-b)\) is not equal to zero, as division by zero is undefined in mathematics.

After performing this step, we obtain \(x = \frac{y}{a-b}\), effectively finding \(x\) in terms of the other variables.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to simplify or solve them. It requires us to apply rules and properties of algebra, such as the distributive property, combining like terms, and inverse operations. In the given exercise \(y = (a-b)x\), you don't necessarily need to combine terms or apply distributive property, but understanding the equation's structure is crucial.
When dividing both sides by \((a-b)\), we used the inverse operation principle to remove the multiplication effect of \((a-b)\) on \(x\).

• Inverse operations like division can help isolate variables effectively.
• It's essential to work through each step with precision, ensuring the equation remains balanced.

Performing these manipulations correctly leads us to the solution \(x = \frac{y}{a-b}\), a simplified form of the original equation, expressing \(x\) solely as a function of \(y\), \(a\), and \(b\).