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Chapter 3: Problem 39

Solve and check each equation. $$0.5 q=6$$

Short Answer

Expert verified
The solution to the equation \(0.5q = 6\) is \(q = 12\). This is checked by substituting the value of 'q' back into the original equation: \(0.5 \cdot 12 = 6\), confirming that the solution is correct.

Step by step solution

01

Identify the given equation and variable

The given equation is \(0.5q = 6\), and we need to find the value of the variable 'q'.
02

Isolate the variable 'q'

To isolate 'q', we will divide both sides of the equation by the coefficient of the variable, which is 0.5. \(q = \frac{6}{0.5}\)
03

Calculate the value of 'q'

Divide 6 by 0.5 to find the value of 'q': \(q = \frac{6}{0.5} = 12\)
04

Check the solution by substituting the value of 'q' back into the original equation

Now we'll check if the value of 'q' we found (12) satisfies the original equation, \(0.5q = 6\): \(0.5 \cdot 12 = 6\) \(6 = 6\) Since both sides of the equation are equal, the solution is correct, and q = 12 is the correct value for the given equation.

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Key Concepts

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

Isolate the Variable
In solving linear equations, one of the most crucial and first steps is isolating the variable you need to find. The goal is to have the variable by itself on one side of the equation. Consider the equation from the exercise, 0.5q = 6. Here, we are solving for 'q'. To isolate 'q', we need to separate it from the coefficient next to it, which in this case, is 0.5. By performing opposite operations, you effectively isolate the variable.
  • Identify the variable and coefficient: In our example, "q" is the variable, and "0.5" is the coefficient.
  • Opposite operation: Since "0.5q" means 0.5 times q, the opposite is division. Divide each side of the equation by 0.5.
This results in: \[q = \frac{6}{0.5}\]By isolating the variable, we transform the equation into a simpler form that reveals the value of 'q'. This process is essential for solving any linear equation effectively.
Verify Solutions
Once the variable has been isolated and a value found, it's essential to verify that the solution is correct. Verifying solutions helps ensure there aren't any mistakes made in calculations. For the given equation, once you calculate that \(q = 12\), it鈥檚 time to confirm that this value is correct.
  • Substitute the variable: Replace 'q' with 12 in the original equation, 0.5q = 6.
  • Simplify: Calculate the left side of the equation to see if it equals the right side.
When this check is done in the initial problem: \[0.5 \times 12 = 6\]This simplifies to \(6 = 6\), confirming that our solution is indeed accurate. Verification is a quick but crucial step to prevent errors and ensure the final result is reliable.
Multiplication and Division in Equations
Working with multiplication and division is a fundamental part of solving linear equations. They help alter the structure of an equation to simplify it. When an equation like \(0.5q = 6\) arises, our task is to handle the multiplication of 0.5 efficiently through division.
  • Identify operations: Notice that 0.5 is being multiplied by 'q', so to isolate 'q', divide both sides by 0.5.
  • Perform division: Calculate the right side by dividing the number by the coefficient to get the solution.
This practice not only isolates the variable but also assists in solving for it precisely without making common errors. \[q = \frac{6}{0.5} = 12\]By mastering multiplication and division in equations, you facilitate smoother transitions through solving processes, making complex equations appear less daunting.

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