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Chapter 8: Problem 235

In the following exercises, solve the equation by clearing the decimals. $$ 3.6 j-2=5.2 $$

Short Answer

Expert verified
j = 2

Step by step solution

01

- Eliminate the Decimals

To clear the decimals, multiply every term in the equation by 10 because the maximum number of decimal places in the terms is 1.Equation: \[3.6 j - 2 = 5.2\]Multiply each term by 10:\[10 \times 3.6 j - 10 \times 2 = 10 \times 5.2\]This simplifies to:\[36 j - 20 = 52\]
02

- Isolate the Variable Term

Add 20 to both sides of the equation to isolate the term with the variable:\[36 j - 20 + 20 = 52 + 20\]This simplifies to:\[36 j = 72\]
03

- Solve for the Variable

Divide both sides by 36 to solve for \( j \):\[\frac{36 j}{36} = \frac{72}{36}\]This simplifies to:\[j = 2\]

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Key Concepts

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

clear decimals in equations
When solving equations, decimals can complicate things. It is often easier first to clear the decimals. This simplifies the equation and makes calculations more straightforward.
To do this, find the number with the most decimal places. In our example, the equation is:
\[3.6 j - 2 = 5.2\]
Here, both terms with decimals have one decimal place. Multiply each term by 10 (because 10 moves the decimal point one place to the right). After multiplying, the equation becomes:
\[36 j - 20 = 52\] \Now, the decimals are gone, and the equation looks simpler.

isolate variable
Once the decimals are cleared, the next step is to isolate the variable. Isolating the variable means getting the variable on one side by itself. In our example, we have:
\[36 j - 20 = 52\]
To isolate \(j\), you add 20 to both sides. This step eliminates the constant term on the left side:
\[36 j - 20 + 20 = 52 + 20\]
And this simplifies to:
\[36 j = 72\]
The variable \(j\) is almost isolated now. The constant term has been moved to the other side.

solve linear equations
After isolating the variable, the final step is solving the linear equation. A linear equation is an equation of the first degree, meaning it has no exponents higher than one. Now, the equation looks like this:
\[36 j = 72\]
To solve for \(j\), divide both sides by 36. This step isolates \(j\) completely:
\[\frac{36 j}{36} = \frac{72}{36}\]
This simplifies to:
\[j = 2\]
Now, we have found the solution: \(j = 2\). Always remember to check your solution by plugging it back into the original equation to ensure correctness.

By mastering these steps鈥攃learing decimals, isolating the variable, and solving the linear equation鈥攜ou can tackle any similar problem effectively.

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