Problem 55 - Introductory Algebra for College Students

Question

Determine whether the given number is a solution of the equation. $$6(p-4)=3 p ; 8$$

Step-by-Step Solution

Answer

Yes, the number 8 is a solution to the equation.

Step 1: Simplify The Equation

Begin with the equation \(6(p-4) = 3p\). Distribute the 6 to both the \( p \) and the -4, to give: \(6p - 24 = 3p\)

Step 2: Isolate the Variable

To isolate \( p \) on one side, subtract \( 3p \) from both sides. This gives: \(3p - 24 = 0\). Then, add 24 on both sides to fully isolate \( p \), which then gives: \(3p = 24\). So \( p = 24 / 3 \), therefore \( p = 8 \)

Step 3: Verify The Solution

To check if 8 is the solution, substitute it back into the original equation. \(6(8-4) = 3*8\), simplified to \(24 = 24\). Both sides match, so 8 is a solution of the equation

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. In the given problem, the expression is \(6(p-4) = 3p\). Here, \(p\) is a variable that represents an unknown number. Algebraic expressions can contain:

• Constants, like numbers \(6\), \(-4\), and \(3\)
• Variables, such as \(p\), which can take different values
• Operators, like addition, subtraction, or multiplication

Expressions are simplified by performing operations. In this case, we use the distributive property to eliminate parentheses. Simplifying \(6(p-4)\) gives \(6p - 24\). Understanding how to manipulate expressions is key in algebra, as it helps to solve equations and determine the relationships between different quantities.

Solving Equations

Solving an equation involves finding the value of the variable that makes the equation true. The equation \(6p - 24 = 3p\) needs to be solved for \(p\). We do this by isolating \(p\) on one side:

• First, subtract \(3p\) from both sides to simplify: \(3p - 24 = 0\)
• Next, add 24 to both sides: \(3p = 24\)
• Finally, divide by 3 to solve for \(p\): \(p = 8\)

Each step has a specific goal: simplifying expressions, combining like terms, and eventually isolating the variable. This process is vital, as it not only finds the solution but also ensures it's systematically reached, making the logical process replicable for any linear equation of similar structure.

Verification of Solution

Verification of a solution ensures that the proposed value satisfies the original equation. After finding \(p = 8\), we check it by returning to the original equation: \(6(p-4) = 3p\) and substituting \(8\) for \(p\).

• Calculate the left side: \(6(8-4) = 6 \times 4 = 24\)
• Calculate the right side: \(3 \times 8 = 24\)

Since both sides equal \(24\), the value \(p = 8\) satisfies the equation. Verification is a critical step as it confirms the correctness of the solution, preventing errors and ensuring that the answer is indeed valid. Always verify your solutions, especially in algebra, to solidify understanding and accuracy.