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Chapter 5: Problem 34

In the following problems, solve each of the conditional equations. $$ 3.06 m=12.546 $$

Short Answer

Expert verified
Question: Solve for the variable m in the given equation: 3.06m = 12.546. Answer: m = 4.1.

Step by step solution

01

Identify the equation

The equation we need to solve is: $$ 3.06m=12.546 $$
02

Isolate the variable

In order to solve for the variable \(m\), we need to isolate it. To do this, we will divide both sides by the coefficient, 3.06. So, we have: $$ \frac{3.06m}{3.06} = \frac{12.546}{3.06} $$
03

Perform the division

Now we need to perform the division on both sides: $$ m = \frac{12.546}{3.06} $$
04

Calculate the value of the variable

Perform the division to obtain the value of \(m\): $$ m = 4.1 $$ The solution for the equation is \(m = 4.1\).

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Key Concepts

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

Variable Isolation Basics
Variable isolation is a fundamental skill in solving linear equations. It means rearranging the equation so that the unknown variable appears by itself on one side of the equation. This helps us understand what the variable equals.

Let's take the equation from our problem:
  • The equation is: \(3.06m = 12.546\).
  • Our goal is to isolate \(m\).
  • The equation tells us that 3.06 times \(m\) equals 12.546.
To isolate \(m\), we need to reverse the operation being performed on it. Since \(m\) is being multiplied by 3.06, we do the opposite operation, which is division. This process will reveal the direct relationship between \(m\) and the numerical value on the other side of the equation.
Understanding the Division Operation
Performing division is a crucial step in solving for a variable, particularly in linear equations. Division is the inverse operation of multiplication and is used here to solve for \(m\).

Here, the division operation is used to isolate \(m\) in the equation \(3.06m = 12.546\):
  • Divide both sides by 3.06 ensures that \(m\) stands alone.
  • When we divide both sides: \(\frac{3.06m}{3.06} = \frac{12.546}{3.06}\).
  • The left side simplifies to \(m\) because \(3.06/3.06 = 1\).
After performing this division, we achieve our goal of having \(m\) isolated by itself on one side. The fraction \(\frac{12.546}{3.06}\) gives us the value of \(m\), simplifying our equation further.
Exploring Conditional Equations
Conditional equations are equations that can be true for some values of the variables involved, but not all. This means solving them is essential to find the specific value(s) that satisfy the equation.

In the equation \(3.06m = 12.546\), \(m\) can take only one particular value to make the equation true. To find this value, systematic steps must be followed:
  • Identify and isolate the variable - This directs us on how to manipulate the equation.
  • Perform necessary operations - Here, that's division to clear the coefficient from \(m\).
By calculating \(\frac{12.546}{3.06}\), we find that \(m = 4.1\), which is the only value that satisfies the original equation. Ensuring precision in these calculations is key in solving conditional equations accurately.

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