Problem 56 - Introductory Algebra for College Students

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 of the equation.

Step 1: Substitute given value

Replace p in the equation \(4(p+3)=6p\) with the given value, 6. This gives the equation \(4(6+3)=6*6\).

Step 2: Simplify both sides

Simplify the equation by performing the operations according to orders of operations. This gives \(4*9 = 36\) on the left side and \(6*6 = 36\) on the right side.

Step 3: Compare the sides

Compare both sides of the equation after calculation. If both sides are equal, then 6 is a solution of the equation. In this case, both sides result in 36, thus 6 is a solution.

Key Concepts

Checking Solutions

Checking if a given number is a solution to an equation means plugging that number into the equation to see if it makes the equation true. This process is a fundamental part of algebra, allowing you to verify your intuition or educated guess.
To begin checking solutions, first take the original equation. For example, with the equation \(4(p+3)=6p\), if you're given the number 6 as a potential solution, you need to substitute 6 wherever you see \(p\) in the equation.
After substitution, the equation turns into \[4(6+3)=6 \times 6\] Simplify both sides to determine if they equal the same value.

• If both sides equal the same number, then 6 is indeed a solution to the equation.
• If not, then 6 is not a solution. The beauty of checking solutions with algebraic equations is that it provides a clear, objective reality—number can either fulfill the equation, or it cannot.

Substitution Method

The substitution method is a technique commonly used in solving algebra equations. This method involves replacing a variable with a given number to see if it satisfies the equation.
In the exercise, we were given \(p = 6\), and we substituted \(6\) for \(p\) in the equation \(4(p+3)=6p\). This substitution effectively turns the equation from an abstract statement into something very concrete:

• Replace \(p\) with 6, giving \(4(6+3) = 6 \times 6\).

Once the substitution is complete, the equation becomes purely numerical, which is much simpler to handle because it no longer contains variables. This guides us directly to the simplification stage. Substituting a variable with a particular value also allows you to test specific scenarios to find the solution to an equation.

Equation Simplification

Equation simplification involves making a complicated equation easier to work with by performing basic arithmetic operations.
Once you've done substitution, the next step is to simplify the equation by carrying out operations:

• First, consider any parentheses in the equation. Add or subtract inside them.
• Next, follow the order of operations, performing multiplication or division.

In our example, substituting \(p = 6\) in \(4(p+3)=6p\) transformed into:\
\(4(6+3)=6 \times 6\). Simplifying this gives:\

• The left side: \(4 \times 9 = 36\)
• The right side: \(6 \times 6 = 36\)

Both simplified expression results are equal, confirming that the given value is a solution.
Simplification helps to double-check our solution, ensuring steps are followed in the correct order, supporting the argument that equations can be resolved through methodical and careful simplification.