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Chapter 3: Problem 30

Solve and check each equation. $$\frac{w}{5}=4$$

Short Answer

Expert verified
The solution to the equation \(\frac{w}{5}=4\) is \(w=20\), as confirmed by checking the solution: \(\frac{20}{5}=4\).

Step by step solution

01

Multiply both sides by 5

First, we need to multiply both sides of the equation by 5 to remove the fraction and isolate w. So, we have: \(5 \cdot \frac{w}{5} = 4 \cdot 5\)
02

Solve for w

Now, we'll simplify the equation: \(w = 20\)
03

Check the solution

To check the solution, insert the value of w back into the original equation: \(\frac{w}{5}=4\) \(\frac{20}{5}=4\) 4 = 4 Since the equation holds true, the solution w = 20 is correct.

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Key Concepts

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

Equation Simplification
When solving linear equations, one of the first steps usually involves simplifying the equation to make it easier to solve. In our example equation \(\frac{w}{5} = 4\), the fraction \(\frac{w}{5}\) makes the equation slightly more complex. Simplifying this involves removing the fraction by using multiplication.

To simplify, we multiply both sides of the equation by 5, which is the denominator of \(w/5\). This step is performed to "clear" the fraction, allowing us to work with a simpler, integer form of the equation. Thus:
  • Multiply both sides: \(5 \cdot \frac{w}{5} = 4 \cdot 5\).
  • This results in: \(w = 20\).
Once the fraction is cleared, the equation becomes much easier to interpret and solve, leading to the direct finding of the variable.
Checking Solutions
After finding a potential solution to an equation, it is critical to confirm its correctness. This step ensures that no mistakes were made during the simplification process. Checking the solution involves substituting it back into the original equation to verify its validity.

In this case, we found \(w = 20\) as a solution. To check:
  • Insert it back into the original equation: \(\frac{w}{5} = 4\).
  • Substitute \(w = 20\) to get: \(\frac{20}{5} = 4\).
Since the left side of the equation equals the right side (4 = 4), the solution is verified as correct. This proves that our previous calculations stood up to scrutiny. Always remember to double-check, as it solidifies understanding and accuracy.
Isolation of Variables
Isolating the variable is a fundamental strategy in solving linear equations. The goal is to get the variable, in this case, \(w\), all by itself on one side of the equation. By doing this, you directly determine the value of the variable.

When we started with \(\frac{w}{5} = 4\), the variable \(w\) was part of a fraction. To isolate \(w\), we used equation simplification techniques by multiplying both sides by 5.

The basic steps and ideas for isolating a variable are:
  • Identify any operations involving the variable (here, division by 5).
  • Use the opposite operation to cancel it out (multiplication by 5 in this case).
  • Ensure that this operation is applied on both sides of the equation to maintain its balance.
Isolating variables is a powerful practice, allowing for straightforward results. Once isolated, the equation simplifies to a basic comparison, leading you directly to the solution.

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Most popular questions from this chapter

Solve using the five "Steps for Solving Applied Problems." See Examples \(7-9\) Four more than a number is fifteen. Find the number.

Solve each equation by first clearing fractions or decimals. $$\frac{a}{8}-1=\frac{a}{3}-\frac{7}{12}$$

Seven more than twice a number is thirty-five. Find the number. Five more than twice a number is fifty-three. Find the number.

Objectives 2 and 3 Mixed Exercises Solve using the five "Steps for Solving Applied Problems." In August 2007 , there were 2600 entering freshmen at a state university. This is 4\% higher than the number of freshmen entering the school in August of 2006 How many freshmen were enrolled in August \(2006 ?\)

Solve. If the exchange rate between the American dollar and the Japanese Yen is such that \(\$ 4.00=442\) yen, how many yen could be exchanged for \(\$ 30.00 ?\)
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