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Chapter 2: Problem 50

Solve each equation. Check each solution. $$ 2+0.4 p=2 $$

Short Answer

Expert verified
The solution is \(p = 0\).

Step by step solution

01

Isolate the variable term

The equation is given as \(2 + 0.4p = 2\). The first step to solving the equation is to isolate the term containing the variable \(p\). We do this by subtracting 2 from both sides of the equation: \ 2 + 0.4p - 2 = 2 - 2 \ This simplifies to \(0.4p = 0\).
02

Solve for the variable

Now that we have \(0.4p = 0\), we can solve for \(p\) by dividing both sides of the equation by 0.4: \ \frac{0.4p}{0.4} = \frac{0}{0.4} \ This gives us \(p = 0\).
03

Check the solution

We check our solution \(p = 0\) by substituting it back into the original equation: \ 2 + 0.4(0) = 2 \ Simplifying the left side, we have \(2 + 0 = 2\). Since both sides of the equation are equal, our solution is correct.

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Key Concepts

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

Isolation of Variables
The first step in solving a linear equation is to focus on isolating the variable. This allows you to identify the true value of the variable in question. In the equation provided, \(2 + 0.4p = 2\), our goal is to single out \(p\). By doing so, we can manipulate the equation until \(p\) stands alone on one side. To achieve this, subtract 2 from both sides to nullify the constant term on the left-hand side.
  • The original equation is \(2 + 0.4p = 2\).
  • Subtract 2 from both sides: \(2 + 0.4p - 2 = 2 - 2\).
  • This results in \(0.4p = 0\).
Once the variable term is isolated, you can easily proceed to solve for \(p\). Isolating the variable is important because it transforms a more complex equation into a simpler one, making it easier to solve.
Checking Solutions
After finding a potential solution to the equation, it is crucial to check its correctness. This step verifies that our calculations are accurate. Once you have determined \(p = 0\), substitute it back into the original equation \(2 + 0.4p = 2\) to ensure both sides remain equal.
  • Start with the solution: \(p = 0\).
  • Substitute back: \(2 + 0.4(0)\).
  • Simplify: \(2 + 0 = 2\).
The equation holds true, confirming that \(p = 0\) is indeed the correct solution. Using this method of checking helps prevent errors and ensures the solution makes sense in the context of the problem. It not only solidifies your understanding of the steps taken but also boosts confidence in your problem-solving abilities.
Algebraic Simplification
Simplification is a powerful tool in algebra that helps make equations easier to work with. In our task, once we isolate the variable, we can simplify the problem by dividing or multiplying terms to solve for the unknown variable. With \(0.4p = 0\), division is the necessary operation.
  • Divide both sides by 0.4 to solve for \(p\): \(\frac{0.4p}{0.4} = \frac{0}{0.4}\).
  • This simplifies to \(p = 0\).
This method removes the coefficient in front of the variable, leaving \(p\) alone. Simplification reduces complex expressions and equations, making them easier to handle and understand. It focuses on maintaining balance within the equation, ensuring what you do to one side, you reciprocate on the other. Thus, simplicity is key in ensuring clarity and accuracy when working with algebraic equations.

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Most popular questions from this chapter

Dennis and Nancy Wood are celebrating their 30 th wedding anniversary by having a reception at Tiffany Oaks reception hall. They have budgeted \(\$ 3000\) for their reception. If the reception hall charges a \(\$ 50.00\) cleanup fee plus \(\$ 34\) per person. find the greatest number of people that they may invite and still stay within their budget.

To "break even "in a manufacturing business, revenue \(R\) (income) must equal the cost \(C\) of production, or \(R=C\). Exercises 41 through 44 involve finding the break-even point for manufacturing. Discuss what happens if a company makes and sells fewer products than the break-even point. Discuss what happens if more products than the break-even point are made and sold.

Let \(x=1\) and then \(x=2\) in the equation \(x+5=x+6 .\) Is either number a solution? How many solutions do you think this equation has? Explain your answer.

Evaluate the following expressions for the given values. See Section 1.7. If \(x=-2,\) find \(x^{3}-x^{2}-x\)

Use a calculator to determine whether the given value is a solution of the given equation. $$8.13+5.85 y=20.05 y-8.91 ; y=1.2$$
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