Problem 46 - Introductory Algebra for College Students

Question

Solve each equation for \(x .\) $$y=(a+b) x-8$$

Step-by-Step Solution

Answer

\(x = \frac{y + 8}{a + b}\)

Step 1: Isolate the x-term on one side

First, we want to isolate the \(x\) term on one side of the equation. To do that, we add 8 to both sides of the equation: \(y + 8 = (a+b)x\).

Step 2: Isolate x

The next step is to isolate \(x\). We can achieve this by dividing both sides of the equation by \(a + b\), as long as \(a + b\) is not equal to 0. Doing so, we get: \(x = \frac{y + 8}{a + b}\).

Key Concepts

Solving Equations

Solving equations is a foundational skill in algebra that allows us to find unknown values. It involves finding the variable that makes the equation true. Equations often use symbols like \(x\) or \(y\) that represent unknown numbers.
To solve an equation like the one given, we first want to ensure that all variable expressions are isolated on one side while constants are gathered on the other side.
This gives us a clear pathway to determine the value of the unknown variable.

• Analyze the equation to identify the terms.
• Use mathematical operations to isolate the variable.

Each step requires methodical adjustments to balance both sides of the equation, adhering to the principle that whatever you do to one side, you must also do to the other.

Isolate Variable

The ultimate target when solving an equation is to isolate the variable we are interested in. In our specific example, we need to isolate the variable \(x\).
Initially, the equation is given as \(y = (a+b)x - 8\). Here, \(x\) is mingled with both a multiplication by \(a+b\) and a subtraction of 8.
The process of isolation involves:

• Removing all elements surrounding the variable.
• Using inverse operations such as addition to cancel out subtraction.

In the example, adding 8 to both sides removes the -8 from the side with \(x\), leading to \(y + 8 = (a+b)x\). This clarifies our equation by centralizing \(x\) with its related terms, paving the way toward final isolation.

Manipulating Equations

Manipulating equations is crucial in algebra. It is how we adjust equations to help isolate the variable or to find a solution more easily.
By manipulating, we mean performing equal operations on both sides of the equation to maintain balance. This can involve:

• Adding or subtracting terms such as moving -8 by adding 8.
• Multiplying or dividing any side by a number, such as dividing by \(a+b\) to finally isolate \(x\).

In the presented exercise, the manipulation begins with addition and is completed with division. These operations are vital since any operation that undoes what's around \(x\) helps us toward the goal of solving the equation.
Always ensure that operations performed do not alter the equation's truth, like dividing by zero, which is undefined.