Problem 12 - Algebra 1: Concepts and Skills

Question

Solve the equation if possible. Determine whether the equation has one solution, no solution, or is an identity. $$ 3(4 c+7)=12 c $$

Step-by-Step Solution

Answer

The equation \(3(4c + 7) = 12c\) has no solution.

Step 1: Simplify the equation

To simplify the equation, distribute the number outside the parentheses to each term inside the parentheses: \(3 * 4c + 3 * 7 = 12c\), which results in \(12c + 21 = 12c\).

Step 2: Isolate the variable c

To isolate the variable c, subtract \(12c\) from both sides of the equation: \(12c + 21 - 12c = 12c - 12c\), which results in \(21 = 0\). This shows the equation has no solution because \(21\) is not equal to \(0\).

Key Concepts

Equation Solving

Equation solving is an essential skill in algebra. The goal is to find the value of the variable that makes the equation true. When solving equations, we follow a systematic approach to isolate the variable and find its value. In the given exercise, we start by simplifying the equation. This is done by using the distributive property. For example, in our given problem, the step involves expanding the expression \(3(4c+7)\), which becomes \(12c + 21\) after distributing the 3.

Once the equation is simplified, the next step is to isolate the variable \(c\). We typically do this by using inverse operations to eliminate other numbers on both sides of the equation, such as addition, subtraction, multiplication, or division. It's crucial to perform the same operation on both sides to maintain equality. However, in our problem, after simplifying, we end up with \(12c + 21 = 12c\). This step reveals further insights that lead us to an unexpected result.

No Solution Equation

Not all equations have a solution. When working through an equation, you might encounter a scenario where the mathematical operations lead to a statement that is logically impossible, such as a non-equal comparison. This is known as a "no solution" equation.

In our particular exercise, after simplifying and attempting to isolate the variable \(c\), we ended up with the statement \(21 = 0\). Clearly, this is a false statement because 21 can never equal 0. When such an outcome occurs, it indicates that the equation has no solution.

• No solution means no value of the variable will satisfy the equation.
• Such equations are also known as contradictions.

Recognizing a no solution equation is crucial because it saves effort in trying to solve an unsolvable problem.

Identity Equation

While the exercise did not result in an identity equation, understanding what it is can be very helpful. An equation is considered an "identity" when every possible value of the variable satisfies the equation. In simpler terms, both sides of the equation are always equal.

For example, consider the equation \(x + 5 = x + 5\). No matter what value you substitute for \(x\), both sides will still be equal, making it an identity equation.

• Identity equations have infinite solutions because any number makes the equation true.
• These types of equations are also known as "always true" equations.

Identifying an identity equation helps differentiate between typical solving processes and conceptual understanding of true mathematical statements. This insight can become particularly useful in more complex algebraic manipulations.