/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 61 Solve and check each equation. $... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 61

Solve and check each equation. $$5-0.4 p=2.6$$

Short Answer

Expert verified
In summary, after isolating the term with the variable and solving the equation, we find that the value of \(p\) that satisfies the equation \(5 - 0.4p = 2.6\) is \(p = 6\).

Step by step solution

01

Isolate the term with the variable

In this case, we want to isolate the term with the variable \(p\), which is \(-0.4p\). To do this, we'll subtract 5 from both sides of the equation: \(-0.4p = 2.6 - 5\)
02

Simplify and solve for the variable

Now we can simplify the right-hand side of the equation: \(-0.4p = -2.4\) Next, we'll solve for the variable \(p\) by dividing both sides of the equation by \(-0.4\): \(\frac{-0.4p}{-0.4} = \frac{-2.4}{-0.4}\) This simplifies to: \(p = 6\)
03

Check the solution

Now that we've found \(p = 6\), we'll substitute this value back into the original equation to check if it's true: \(5 - 0.4(6) = 2.6\) Calculate the left-hand side of the equation: \(5 - 2.4 = 2.6\) This simplifies to: \(2.6 = 2.6\) Since both sides are equal, the solution is correct. Therefore, the value of \(p\) that satisfies the equation is \(p = 6\).

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Key Concepts

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

Isolating the Variable
Isolating the variable is an essential first step when solving linear equations. It allows us to find the value of the variable that makes the equation true. In our given equation, \[5 - 0.4p = 2.6\]the goal is to get the term with the variable, \(-0.4p\),on one side of the equation by itself. This process begins with removing any constants or other terms on the same side as the variable.
  • First, subtract \(5\) from both sides to remove it from the left side: \(-0.4p = 2.6 - 5\).
  • This step results in \(-0.4p = -2.4\). The variable is now isolated, allowing further steps to solve the equation.
Understanding this process helps to clearly see how managing both sides of an equation equally maintains balance and achieves isolation, which simplifies what remains to an easy to handle operation.
Checking the Solution
Once you think you've found the correct solution, it is important to verify that your answer satisfies the original equation. Checking the solution ensures that no mistakes were made during the algebraic process and that the correct mathematical balance is achieved.To check the solution for \(p = 6\), substitute this value back into the original equation:
  • Replace \(p\) with \(6\): \(5 - 0.4(6) = 2.6\).
  • Calculate the left-hand side: \(5 - 2.4 = 2.6\).
The left side simplifies to \(2.6\), which equals the right side. This equality confirms that the proposed value of \(p\) is indeed the solution that satisfies the equation. Checking solutions is an important habit to develop to ensure accuracy in algebra.
Algebraic Simplification
Algebraic simplification involves reducing complex expressions or equations to simpler forms while maintaining their original value. This skill is crucial in solving equations efficiently.For the equation \(-0.4p = -2.4\), simplification comes into play after isolation:
  • Divide both sides by \(-0.4\) to solve for \(p\): \(\frac{-0.4p}{-0.4} = \frac{-2.4}{-0.4}\).
  • By simplifying each side, you find \(p = 6\). This shows how operations like division help eliminate coefficients attached to the variable.
Simplification not only makes equations easier to handle but also clarifies the relationships between components within an equation. It enables a straightforward path to identifying the solution, demonstrating the power of basic algebraic manipulation.

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