Problem 44 - Algebra 1: Concepts and Skills

Question

Check to see if the given value of the variable is or is not a solution of the equation. \(15+2 c=5 c ; c=5\)

Step-by-Step Solution

Answer

Yes, \(c=5\) is a solution to the equation \(15+2c = 5c\).

Step 1: Identify the given equation and value of the variable

The given equation is \(15+2c = 5c\), and the given value of the variable \(c\) is 5. The goal is to check if this given value for \(c\) satisfies the equation.

Step 2: Substitute the value into the equation

Substituting the given value \(c=5\) into the equation gives \(15+2(5) = 5(5)\). This simplifies to \(15+10 = 25\).

Step 3: Verify if the equation holds true

After substituting the value of \(c\) into the equation, it becomes \(25 = 25\). Since both sides of the equation are equal, the given value of \(c\) is indeed a solution to the equation.

Key Concepts

Variable Substitution

Variable substitution is a foundational technique in algebra that involves replacing a variable with a specific value or expression. In the context of solving equations, it serves as a method for determining if a particular value is a solution to the equation.

For example, let's consider the equation \(15+2c=5c\). To determine if \(c=5\) is a solution, we perform a substitution by replacing each occurrence of \(c\) with 5. After substitution, we calculate to see if the equation balances:

\(15 + 2 \times 5 = 5 \times 5\)
\(15 + 10 = 25\)
\

Equation Solutions

Understanding the concept of equation solutions is crucial for students tackling algebraic problems. A solution to an equation is a value that, when substituted for the variable, makes the equation true — meaning both sides of the equation are equal.

In the exercise \(15+2c=5c\), we are checking if \(c=5\) is a solution. We replace \(c\) with 5 and simplify the expression. If the left and right sides of the equation are equal after substitution, then \(c=5\) is indeed a solution to the equation. This is exactly what is observed in this case, confirming that \(c=5\) satisfies the provided equation. It's like finding the right key for a lock; if it fits and turns, you've found your match.

An equation might have one solution, multiple solutions, or sometimes no solution at all. Identifying the correct solution(s) or recognizing there are none is a key skill in algebra.

Algebraic Verification

Algebraic verification is the 'proof' that shows us whether a variable substitution yields the true solution to an equation. The process involves substituting the proposed solution back into the original equation and simplifying the expression to see if the equation holds up.

In the given exercise, we are verifying \(c=5\) for the equation \(15+2c=5c\). The algebraic verification process involves these steps:

• Substituting \(c=5\) into the equation
• Simplifying the equation after substitution
• Comparing both sides of the equation for equality

After performing these steps, we observe \(25 = 25\), confirming that both sides of the equation are equal, thus verifying the solution. Through algebraic verification, not only do we confirm the validity of the solution, but we also understand the 'reasonability' of our algebraic manipulations and establish confidence in our solution.