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Chapter 2: Problem 96

Solve for \(x .\) Assume \(a, c, m \neq 0\) $$ a x-3 a=5 a $$

Short Answer

Expert verified
x = 8

Step by step solution

01

- Combine like terms

Start by noting the terms that contain the variable and constants separately. Rewrite the equation grouping the terms that contain the variable on one side:a x - 3 a = 5 a Since both sides of the equation are divisible by 'a', simplify by dividing both sides by 'a'.
02

- Simplify the equation

Divide every term by 'a':a x / a - 3 a / a = 5 a / a This simplifies to:x - 3 = 5
03

- Solve for x

Add 3 to both sides of the equation to isolate x:x - 3 + 3 = 5 + 3 Simplifying the right-hand side gives:x = 8

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
Combining like terms is a crucial step in solving linear equations. Like terms are terms that have the same variable raised to the same power. In the equation given, we have \(ax - 3a = 5a\). To combine like terms, identify terms with the same variable. In this case, \(ax\) is the term with the variable \(x\), and \(-3a\) and \(5a\) are constant terms (since they don鈥檛 involve \(x\)). By grouping the like terms together, we get:

\(ax = 3a + 5a\).

This step prepares the equation for further simplification and ensures that similar terms are managed together, making the problem easier to solve.
Simplifying Equations
The main goal of simplifying equations is to make the equation as straightforward as possible. After combining like terms, the equation \(ax - 3a = 5a\) should be simplified. Here's how we do it:

First, we can note that each term involves the variable \(a\). We can simplify this by dividing every term by \(a\):

\(\frac{ax}{a} - \frac{3a}{a} = \frac{5a}{a}\)

This simplifies to:

\(x - 3 = 5\).

Dividing by \(a\) eliminated the constant coefficient, reducing the equation to a more manageable form. Simplification helps in focusing on the core parts of the equation that directly contribute to solving for the variable.
Isolating Variables
Isolating the variable is the final critical step in solving linear equations. Once the equation is simplified to \(x - 3 = 5\), we need the variable \(x\) to be alone on one side of the equation. Here's how we isolate \(x\):

Add 3 to both sides of the equation:
\(x - 3 + 3 = 5 + 3\)

This simplifies to:

\(x = 8\).

By adding 3, we eliminate the constant term on the left side, thus leaving \(x\) completely isolated. This step transforms the equation into its simplest form, clearly showing the value of the variable.

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