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Chapter 2: Problem 9

In \(3-20,\) solve each equation and check. $$ 0.4 x+8=0.5 x $$

Short Answer

Expert verified
The solution to the equation is \( x = 80 \).

Step by step solution

01

Isolate Variable Terms

First, we want to get all terms involving the variable on one side of the equation and constant terms on the other. Start by subtracting \(0.4x\) from both sides of the equation:\[0.4x + 8 - 0.4x = 0.5x - 0.4x\]This simplifies to:\[8 = 0.1x\]
02

Solve for the Variable

Now that we have \(0.1x\) isolated, divide both sides by 0.1 to solve for \(x\):\[\frac{8}{0.1} = x\]Thus, \(x = 80\).
03

Verify the Solution

Substitute \(x = 80\) back into the original equation to check the solution:\[0.4(80) + 8 = 0.5(80)\]Calculate each side:\[32 + 8 = 40 \,\, \text{and} \,\, 0.5 imes 80 = 40\]Both sides are equal, confirming that \(x = 80\) is the correct solution.

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Key Concepts

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

Isolating Variable Terms
When solving linear equations, the first step is to have all terms containing the variable on one side of the equation. This makes it easier to work with the equation and find the value of the variable. In the problem \(0.4x + 8 = 0.5x\), the variable \(x\) appears on both sides of the equation. Our goal is to "isolate" the terms with \(x\).To do this, we subtract \(0.4x\) from both sides of the equation. This results in:
  • Left side: \(0.4x + 8 - 0.4x = 8\)
  • Right side: \(0.5x - 0.4x = 0.1x\)
Thus, the equation simplifies to \(8 = 0.1x\). Now, the variable terms are isolated, and the next step can be taken to solve this simplified equation.
Solving for Variables
With the variable term isolated on one side of the equation, the next step is figuring out the value of the variable. We have the equation \(8 = 0.1x\). To solve for \(x\), we need to perform operations to make \(x\) stand alone. This involves dividing both sides of the equation by \(0.1\) to "solve" for \(x\):\[\frac{8}{0.1} = x\]Calculating this, we find that \(x = 80\). This step effectively turns the complex equation into a simple statement of value, making it clear what \(x\) represents. Each operations step reduces the complexity, helping to keep track of mathematical manipulations.
Equation Verification
After solving for the variable, the final step is to verify the solution. This ensures that our calculations are correct, and the value found for \(x\) satisfies the original equation. Let's substitute \(x = 80\) back into the original equation:\[0.4(80) + 8 = 0.5(80)\]Calculating separately:
  • \(0.4 \times 80 = 32\)
  • Adding \(32 + 8 = 40\)
  • On the other side, \(0.5 \times 80 = 40\)
Both sides equal 40, confirming the equation holds true with \(x = 80\). This step is crucial as it acts as a check against any possible errors made during problem-solving, ensuring reliability in the solution.

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Most popular questions from this chapter

In \(3-12,\) multiply and express each product in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{b+1}{4} \cdot \frac{12}{5 b+5} $$

In \(25-30,\) perform the indicated operations and write the result in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \left(x^{2}-2 x+1\right) \div \frac{x-1}{3} \cdot \frac{x+4}{3 x} $$

In \(13-24,\) divide and express each quotient in simplest form. In each case, list any values of the variables for which the fractions are not defined. $$ \frac{3}{4} \div \frac{9}{20} $$

In \(3-20,\) solve each equation and check. $$ \frac{7}{2 x-3}=\frac{4}{x} $$

Anthony rode his bicycle to his friend's house, a distance of 1 mile. Then his friend's mother drove them to school, a distance of 12 miles. His friend's mother drove at a rate that is 25 miles per hour faster than Anthony rides his bike. If it took Anthony \(\frac{3}{5}\) of an hour to get to school, at what average rate does he ride his bicycle? (Use distance \(=\) time for each part of the trip to school.)
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