Problem 72 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ 6 c=-96 $$

Step-by-Step Solution

Answer

The solution to the equation \(6c = -96\) is \(c = -16\).

Step 1: Identify the equation

First, it is clear the equation to be solved is \(6c = -96\). The aim is to find the value of 'c' which makes the equation true.

Step 2: Isolate 'c'

To solve for 'c', it must be isolated on one side of the equation. Since 'c' is currently being multiplied by 6, the next step is to divide both sides of the equation by 6. \(c = -96/6\)

Step 3: Solve for 'c'

Now, simply compute the division. \(c = -16\).

Key Concepts

Isolate Variables

When faced with an algebraic equation like \(6c = -96\), the main goal is to find the value of the variable, in this case, 'c'. To do this, one must 'isolate' the variable. Isolation means to get the variable by itself on one side of the equation, separating it from any other numbers or variables. Here, 'c' is multiplied by 6, hence our task is to reverse this operation. How do we do that? We perform the inverse algebraic operation, which is division in this case. Dividing both sides of the equation by 6 undoes the multiplication and results in 'c' being by itself, leading to a simpler equation, \(c = -96/6\).

Isolation is a fundamental skill in algebra as it applies across various types of equations, from simple linear equations to more complex polynomial equations. Mastering this technique is key to understanding and solving algebraic expressions effectively.

Algebraic Operations

Algebraic operations are the building blocks of solving equations. They consist of addition, subtraction, multiplication, and division. But it does not end there; there are also more advanced operations like exponentiation and root extraction. In the context of the given exercise, we are looking at multiplication and division as the primary operations. For the equation \(6c = -96\), we use multiplication to understand how 'c' is related to -96, and then we use division to reverse this operation and solve for 'c'.

To deploy algebraic operations correctly, one must adhere to the rules of algebra, such as the distribution of multiplication over addition and the associative and commutative properties. By using inverse operations to cancel out terms, we manipulate the equation step by step until only the desired variable remains on one side of the equal sign.

Division in Algebra

Division is the process of determining how many times one number is contained within another, and in algebra, it is used to undo multiplication. When we have an equation like \(6c = -96\), division is the key to isolating 'c'. By dividing both sides of the equation by 6 (the number that 'c' is multiplied by), we determine the value of 'c' that satisfies the equation. Here, division simplifies the equation to its core, showing that \(c = -16\). It's important to divide both sides of the equation by the same number to maintain balance and keep the equation true.

Remember that division by zero is undefined in algebra, which emphasizes the need to recognize restrictions on variables when solving equations. Understanding division helps in breaking down complex relationships between numbers and variables and is an essential part of problem solving in algebra.