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Chapter 2: Problem 81

\(13 p+3(2-4 p)=4\)

Short Answer

Expert verified
The solution is p = -2.

Step by step solution

01

- Distribute

First, distribute the 3 through the parentheses: 3(2 - 4p) becomes 6 - 12p. Now the equation is: 13p + 6 - 12p = 4.
02

- Combine Like Terms

Next, combine the like terms on the left side of the equation: 13p - 12p = p. So the equation simplifies to: p + 6 = 4.
03

- Solve for p

Subtract 6 from both sides to isolate p: p + 6 - 6 = 4 - 6 p = -2.

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Key Concepts

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

Distributive Property
The distributive property is a fundamental principle in algebra. It states that multiplying a number by a sum is the same as multiplying the number by each addend and then adding the products. For example, suppose we have the equation: 3(2 - 4p).
To apply the distributive property:
  • Multiply 3 by 2, which gives us 6.
  • Then, multiply 3 by -4p, which gives us -12p.
So, 3(2 - 4p) simplifies to 6 - 12p. This way, we can eliminate the parentheses and make it easier to solve the equation.
Combining Like Terms
Combining like terms is another essential step in solving linear equations. Like terms are terms that contain the same variable raised to the same power. For instance, in the equation 13p + 6 - 12p = 4:
  • The terms 13p and -12p are like terms because they both contain the variable 'p'.
To combine them, you simply add or subtract their coefficients:
  • 13p - 12p equals p.
Thus, the equation simplifies to p + 6 = 4. Combining like terms helps in reducing the equation to a simpler form, making it easier to isolate the variable and solve the problem.
Isolating Variables
Isolating the variable is a crucial step in solving equations. It means getting the variable by itself on one side of the equation. From the simplified equation p + 6 = 4:
  • Subtract 6 from both sides to keep the equation balanced: p + 6 - 6 = 4 - 6.
This simplifies to p = -2. Now the variable 'p' is isolated, and we have our solution. Isolating the variable often involves inverse operations like addition and subtraction, ensuring we keep the equation balanced while simplifying it further.

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