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Chapter 3: Problem 65

Solve the equation. $$3 p+4 p=25-4$$

Short Answer

Expert verified
p = 3

Step by step solution

01

Combine Like Terms

Combine the terms on the left side that contain the variable. In this case, combine \(3p + 4p\): 3p + 4p = 7p, so the equation simplifies to: 7p = 25 - 4.
02

Simplify the Right Side

Simplify the right side of the equation by performing the subtraction: 25 - 4 = 21, so the equation simplifies to: 7p = 21.
03

Solve for the Variable

Isolate the variable by dividing both sides of the equation by 7: \[ p = \frac{21}{7} \]. Perform the division: p = 3.

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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 a crucial skill in solving linear equations. When we say 'like terms', we mean terms that have the exact same variable part. For instance, in the expression \(3p + 4p\), both terms contain the variable \(p\), making them 'like terms'.

To combine them, simply add their coefficients (the numbers in front of the variables). Here, you would add 3 and 4 to get 7. So, \(3p + 4p\) becomes \(7p\).

Remember:
  • Like terms share the same variables and exponents.
  • Combine them by adding or subtracting their coefficients.
Combining like terms simplifies the equation and makes it easier to solve.
Simplifying Expressions
Simplifying expressions means breaking down complex mathematical expressions into simpler forms. In our equation, \(3p + 4p = 25 - 4\), after combining like terms we get \(7p = 25 - 4\).

Now, simplify the right side by performing the subtraction: \(25 - 4 = 21\). This reduces our equation to \(7p = 21\).

To simplify expressions effectively:
  • Perform all straightforward arithmetic operations.
  • Combine like terms whenever possible.
Such steps help reduce the number of terms in the equation, making it simpler and easier to work with.
Isolating the Variable
Isolating the variable is the process of manipulating an equation to get the variable alone on one side of the equation. In our simplified equation \(7p = 21\), our goal is to solve for \(p\) by isolating it.

To achieve this, we need to get rid of the coefficient 7 that is multiplied by \(p\). We do this by performing the inverse operation. Since 7 is multiplying \(p\), we divide both sides of the equation by 7:

\[\begin{equation} p = \frac{21}{7} \end{equation}\]

After performing the division, we find \(p = 3\).

Key steps for isolating the variable:
  • Identify the variable you need to isolate.
  • Use inverse operations (addition/subtraction or multiplication/division) to move other terms across the equation.
  • Simplify to solve for the variable.
This step is essential for finding the solution to the equation.

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