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Chapter 1: Problem 63

Solve each equation. \(4.2 x+6.4=40\)

Short Answer

Expert verified
The solution to the equation is \(x = 8\).

Step by step solution

01

Isolate the variable term

Start with the equation: \(4.2x + 6.4 = 40\). To isolate the term with \(x\), subtract \(6.4\) from both sides of the equation. This gives: \(4.2x = 40 - 6.4\).
02

Simplify the equation

Calculate \(40 - 6.4\). The result is \(33.6\). Thus, the equation simplifies to \(4.2x = 33.6\).
03

Solve for x

To solve for \(x\), divide both sides of the equation by \(4.2\). This gives: \(x = \frac{33.6}{4.2}\).
04

Calculate the value of x

Perform the division: \(x = 8\). Hence, the solution to the equation is \(x = 8\).

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Key Concepts

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

Isolation of Variable
In mathematical equations, isolating the variable is like zooming in on the main character of a story. Imagine you have a balance scale, and you want to know the weight of an unknown object. You put it on one side and gradually remove known weights from both sides until you're left with just the object. This is the essence of isolating a variable.

For example, in the equation \(4.2x + 6.4 = 40\), our goal is to isolate \(x\). To do this, we need to remove the 'extra' number \(6.4\) that is currently added to \(4.2x\).
  • Subtract \(6.4\) from both sides of the equation, which keeps the equation balanced, much like balancing both sides of a scale.
  • After this step, the equation simplifies to \(4.2x = 33.6\).
Now, the term with \(x\) stands alone on one side, ready for further simplification.
Simplification
After isolating the variable's term, it's time to simplify the equation so it looks neater and more straightforward to solve. Simplification is like cleaning up a messy desk, so it's easier to focus on the task at hand.

From our previous step, we have the equation \(4.2x = 33.6\). To make it simpler, we perform the subtraction \(40 - 6.4\), transforming our equation to a cleaner form where \(4.2x = 33.6\).
  • Simplification removes any additional numbers, making the process clearer.
  • This makes calculating the final result much simpler.
With this simplification, the journey to solving for \(x\) is straightforward from here.
Division in Equations
Division is a powerful tool in solving linear equations and is used to make unknown variables easy to find. Once your equation is simplified and the variable is isolated, division is often the final step in solving for that variable.

In our example, after simplifying to \(4.2x = 33.6\), we need to completely isolate \(x\) by dividing both sides of the equation by \(4.2\).
  • This step neutralizes the multiplication between \(x\) and \(4.2\).
  • Performing this division, \(\frac{33.6}{4.2}\), gives us \(x = 8\).
By dividing, we find the precise value of the unknown variable, and thus solve the equation. Division breaks down the problem into a solvable part, revealing the solution clearly.

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