Problem 44 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(43-50,\) solve each equation for \(x .\) $$y=(a-b) x$$

Step-by-Step Solution

Answer

The solution for \(x\) is \( x = y / (a-b)\)

Step 1: Identify the equation

The given equation is \( y = (a-b)x \)

Step 2: Isolate x

To solve for \(x\), we need to isolate \(x\). This can be achieved by dividing both sides of the equation by \((a-b)\). This gives us \( x = y / (a-b)\).

Key Concepts

Solving for x

When tasked with solving for \(x\) in an equation, the primary goal is to express \(x\) in terms of the other variables and constants. This process allows us to determine the value of \(x\) when we know the values of the other components in the equation. In the example given, we are working with the equation \(y = (a-b)x\). Here, \(x\) is multiplied by the term \((a-b)\), making it not immediately isolated or alone on one side of the equation. By solving it, we aim to convert this equation into a more manageable form where \(x\) stands alone.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying equations using various operations such as addition, subtraction, multiplication, and division. These operations help in transforming the equations into more useful forms. To manipulate the equation \(y = (a-b)x\), you need to focus on reversing the operations that prevent \(x\) from being isolated. Here, the operation of multiplication is the barrier.

• Reversing Multiplication: In our equation, \(x\) is multiplied by \((a-b)\). To undo this, divide both sides of the equation by \((a-b)\).
• This division cancels out the multiplication, allowing us to isolate \(x\) on one side of the equation.

As you perform these manipulations, always abide by the rule of maintaining equality, meaning whatever operation is performed on one side must equally be performed on the other side as well.

Isolating Variables

Isolating a variable means rearranging an equation so that the variable interest is by itself on one side of the equation. This is an essential skill in algebra, allowing us to solve equations and understand relationships between variables more clearly. In our equation example, \(y = (a-b)x\), \(x\) needs to be isolated to solve the equation for \(x\).

• Start by Identifying: Clearly identify the target variable, which in this case is \(x\).
• Use Algebraic Operations: As \(x\) was initially part of a multiplication with \((a-b)\), you eliminate this by dividing both sides: \(x = \frac{y}{a-b}\).

By conducting these operations correctly, the variable \(x\) becomes separated from any other terms or coefficients, making it easier to interpret its relationship with other variables through its isolated form. This approach is fundamental in understanding and solving many real-world problems where determining the value of an unknown is crucial.