Problem 11 - Prealgebra

Question

Solve each equation. Check your solution. $$4(f+3)+5=17+4 f$$

Step-by-Step Solution

Answer

The solution is all real numbers.

Step 1: Distribute the 4

The equation given is \(4(f+3)+5=17+4f\). First, apply the distributive property to \(4(f+3)\) to obtain \(4f + 12 + 5 = 17 + 4f\).

Step 2: Simplify the Equation

Combine like terms on the left side. The terms \(12\) and \(5\) are constants that can be added together: \(4f + 17 = 17 + 4f\).

Step 3: Eliminate Common Terms

Subtract \(4f\) from both sides of the equation to remove the variable term from one side: \(17 = 17\).

Step 4: Analyze the Result

Note that the simplified result \(17 = 17\) is a true statement. This means that the original equation is always true, regardless of the value of \(f\).

Step 5: State the Solution

The equation is true for all real numbers. Thus, the solution is that any value of \(f\) will satisfy the equation.

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows you to multiply a single term with a group of terms inside parentheses. It's a handy tool to simplify expressions and is often used to clear out parentheses in an equation. This property states that for any numbers or variables a, b, and c, the expression \( a(b + c) \) is equivalent to \( ab + ac \).
For example, in the original equation \(4(f+3) + 5 = 17 + 4f\), we apply the distributive property to \( 4(f+3) \). Here's how it works:

• Multiply 4 by each term inside the parentheses: \(4 \times f\) and \(4 \times 3\).
• This results in \(4f + 12\).

Using the distributive property helps to break down complex equations into simpler parts, making it easier to manipulate and solve them.

Like Terms

In algebra, combining like terms is a crucial step in simplifying expressions and equations. Like terms are terms that have the same variables raised to the same power. Only the coefficients (the numbers in front of the variables) of like terms can be added or subtracted.
For instance, in the expression \(4f + 12 + 5\), notice that \(4f\) is a term with the variable \(f\), while 12 and 5 are constant terms. The like terms here are 12 and 5, as they are both constant numbers without variables.

• To combine, simply add the coefficients: \(12 + 5 = 17\).

By combining like terms, we simplify the expression, making it easier to see what's happening with the variables and constants in your equation.

Equation Simplification

Simplifying equations is the process of reducing them to their simplest form, making it easier to solve or understand. After applying the distributive property and combining like terms, the next step is often to move terms around to isolate variables or eliminate redundancies.
In the exercise, we simplified the equation down to \(4f + 17 = 17 + 4f\). The goal here is to get all variable terms on one side and constants on the other. When we subtract \(4f\) from both sides, what remains is the equation \(17 = 17\).

• Notice this tells us something important: the equation holds true regardless of the value of \(f\).
• This means every real number is a solution, as the equation is inherently balanced.

Equation simplification helps reveal underlying truths about equations, such as whether they are always true, never true, or true for specific values.