Problem 56 - Introductory and Intermediate Algebra for College Students 4th

Question

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

Step-by-Step Solution

Answer

Yes, 6 is a solution to the equation.

Step 1: Substitute Given Number

First, replace the variable \(p\) in the equation \(4(p+3) = 6p\) with the given number \(6\). This gives the equation \(4(6+3) = 6*6\).

Step 2: Simplify the Expressions

In this step, simplify both sides of the equation. That involves first simplifying the operations in the brackets which is \(4*9 =36\) on the left side and \(6*6=36\) on the right side. This results in the equation, \(36=36\).

Step 3: Evaluate the Simplified Statement

Looking at the simplified equation, \(36=36\), it can be seen that both sides of the equation are equal. This is a true statement, indicating that the given number \(6\) is indeed a solution for the equation in question.

Key Concepts

Equation Solving

Equation solving is a fundamental part of algebra where you find the value of the variable that makes the equation true. In our problem, the equation is \(4(p+3) = 6p\). The goal is to determine if the number 6 is a solution for this equation. By solution, we mean that when the number 6 is substituted in place of \(p\), both sides of the equation should be equal.

This exercise tests our understanding of how to manipulate and verify equations by substituting potential solutions. Ensuring this equality verifies whether a chosen value indeed solves the equation.

Substitution Method

The substitution method involves substituting a given value into an equation to check if it holds true. Here, we substituted \(p = 6\) into the equation \(4(p+3) = 6p\). This step is crucial as it transforms the original equation into a numerical form that can be simplified and evaluated for truth.

The substitution process is simple yet powerful, allowing us to test potential solutions directly. By replacing \(p\) with 6, we simplify the problem from dealing with variables to dealing with plain numbers, making it more straightforward to solve.

Simplification

Simplification is about reducing an equation or mathematical expression to its most basic form. After substituting \(p = 6\), our equation becomes \(4(6+3) = 6 \times 6\). The next step is simplifying these expressions.

First, apply the operations inside the parentheses: \(4 \times 9\) on the left, and on the right, perform the multiplication \(6 \times 6\). This simplifies both sides to 36, yielding the equation \(36 = 36\).

By reducing equations to simpler forms, we avoid errors and make verification clear and easy. Proper simplification is key to ensuring accurate solutions.

Mathematical Verification

Mathematical verification involves double-checking calculations to confirm that a substituted value satisfies an equation. In our problem, after simplification, we arrived at \(36 = 36\).

Seeing that both sides of the equation are equal, we can conclude that the number 6 is indeed a correct solution. This validation stage is critical in mathematics to ensure that solutions are not only calculated but verified as accurate.

It's all about instilling confidence that our solution meets all the requirements set by the equation. It helps to ensure there were no calculation errors during substitution or simplification.