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Chapter 2: Problem 118

Solve each equation requiring simplification. $$0.05 p-0.01 p=2+0.24$$

Short Answer

Expert verified
p = 56

Step by step solution

01

Combine Like Terms on the Left Side

Combine the terms involving 'p' on the left-hand side of the equation. Start with the given equation:\[0.05p - 0.01p = 2 + 0.24\]Combining the like terms gives us:\[(0.05 - 0.01)p = 2 + 0.24\]This simplifies to:\[0.04p = 2 + 0.24\]
02

Combine Constants on the Right Side

Now, simplify the right-hand side of the equation by combining the constants:\[0.04p = 2 + 0.24\]Adding the constants gives:\[0.04p = 2.24\]
03

Solve For p

Isolate 'p' by dividing both sides of the equation by 0.04:\[p = \frac{2.24}{0.04}\]Perform the division to find the value of 'p':\[p = 56\]

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Key Concepts

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

Combining Like Terms
Combining like terms is an essential step in solving linear equations. Like terms have the same variable raised to the same power. For example, in the expression \(0.05p - 0.01p\), both terms have the variable \(p\). To simplify, you subtract the coefficients (the numbers in front of \(p\)). Here, we combine the terms as follows:
  • Start with \(0.05p - 0.01p\).
  • Subtract the coefficients: \(0.05 - 0.01 = 0.04\).
  • Simplify to get \(0.04p\).
Combining like terms helps to reduce the equation to a simpler form, making it easier to solve.
Constant Terms
Constant terms are numbers without variables. They remain fixed, and unlike variables, they don't change value. In our example, we have the constants '2' and '0.24' on the right-hand side. To solve the equation, it's crucial to combine these constant terms:
  • We started with \(2 + 0.24\).
  • Add the constants: \(2 + 0.24 = 2.24\).
This step simplifies the right-hand side of the equation. Managing constant terms correctly ensures the equation is solvable and makes it easier to isolate the variable.
Isolating Variables
Isolating the variable is the final and crucial step in solving the equation. The goal is to get the variable on one side of the equation and the constants on the other. In the example, our equation was reduced to \(0.04p = 2.24\). To isolate \(p\):
  • We divided both sides by 0.04.
  • This gives us \(p = \frac{2.24}{0.04}\).
  • Perform the division: \(p = 56\).
This step reveals the value of the variable. Isolating the variable is essential, as it provides the solution to the equation.
Elementary Algebra
Understanding elementary algebra is foundational for solving any algebraic problems. It involves manipulating equations, combining like terms, managing constants, and isolating variables. In our example, we used all these skills to find \(p\). With elementary algebra, you will use:
  • Basic arithmetic operations (addition, subtraction, multiplication, division).
  • Combined like terms to simplify equations.
  • Isolated variables to solve the equations.
Mastering these skills makes solving linear equations straightforward, helping you progress to more complex algebraic topics.

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