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Chapter 4: Problem 49

Solve. $$ 4.2 \cdot x=39.06 $$

Short Answer

Expert verified
x = 9.3

Step by step solution

01

- Understand the Equation

The equation given is a linear equation in the form of \( ax = b \), where \( a = 4.2 \) and \( b = 39.06 \).
02

- Isolate x on One Side

To isolate \( x \), divide both sides of the equation by the coefficient of \( x \) which is 4.2. The equation becomes: \[ x = \frac{39.06}{4.2} \]
03

- Perform the Division

Calculate the division \( \frac{39.06}{4.2} \). This can be done using a calculator or by performing long division: \[ x = 9.3 \]
04

- Verify the Solution

To verify, substitute \( x = 9.3 \) back into the original equation: \[ 4.2 \cdot 9.3 = 39.06 \]. Since both sides are equal, the solution is correct.

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Key Concepts

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

Isolating Variables
In algebra, isolating the variable means getting the variable by itself on one side of the equation. This is a crucial first step to solving linear equations. For instance, in the equation \(4.2 \cdot \ x = 39.06\), the goal is to move everything except \(x\) to the other side. This is typically achieved by performing operations that 'undo' what is being done to the variable. In this case, since \(4.2 \cdot x\) represents multiplication, we undo it by dividing both sides of the equation by 4.2.
Division in Algebra
When solving equations, division is a common operation used to isolate the variable. For example, with \(4.2 \cdot x = 39.06\), you divide both sides by 4.2 to get \(x = \frac{39.06}{4.2}\). The concept here is straightforward - whatever operation you perform on one side of the equation, you must do the same on the other side to maintain equality. Therefore, \( \frac{39.06}{4.2} = 9.3 \). Calculating this either by a calculator or long division will solve for \(x\).
Equation Verification
Once you have isolated the variable and performed the necessary operations, it is always important to verify your solution. Substitution is the most common method to verify solutions. In our case, after finding \(x = 9.3\), you substitute it back into the original equation to ensure both sides are equal. So, \(4.2 \cdot 9.3 = 39.06\). Since the original equation is satisfied, \(x = 9.3\) is the correct solution. Verification prevents errors and confirms that your solution is accurate.

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