Problem 35 - College Algebra

Question

Solve for the variable. $$ 2(11 c-4)=36 $$

Step-by-Step Solution

Answer

The solution is \(c = 2\).

Step 1: Distribute the 2

First, distribute the 2 across the terms inside the parentheses on the left-hand side of the equation. This means you'll multiply 2 by both 11c and -4.\[2(11c - 4) = 36 \22c - 8 = 36\]

Step 2: Add 8 to Both Sides

Next, to isolate the term with the variable on one side, add 8 to both sides of the equation. This cancels out the -8 on the left-hand side.\[22c - 8 + 8 = 36 + 8 \22c = 44\]

Step 3: Divide by 22

Finally, divide both sides of the equation by 22 to solve for the variable \(c\).\[\frac{22c}{22} = \frac{44}{22} \c = 2\]

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra. It involves finding the value of a variable that makes the equation true. When you are solving equations, your main goal is to isolate the variable, usually represented by a letter like \( c \), on one side of the equation. This allows you to clearly see its value.

• Start by simplifying both sides of the equation if necessary.
• Use inverse operations to move other numbers to the opposite side.
• Perform the same operation on both sides to maintain balance.

Understanding this process is key to tackling more complex math problems. It's like cooking, where each step needs to be done in order. If done wrong, the recipe, or in this case, the problem, won't turn out right.

Distribution Property

The distribution property is a powerful tool in algebra. It allows you to simplify expressions by distributing a factor across terms inside parentheses. In the equation \( 2(11c - 4) = 36 \), you apply the distribution property by multiplying the \( 2 \) by each term inside the parentheses. This means:

• \( 2 \times 11c = 22c \)
• \( 2 \times -4 = -8 \)

Once applied, the equation becomes \( 22c - 8 = 36 \). This step simplifies the equation and brings us closer to isolating the variable. Think of distribution as breaking down a complex task into smaller, manageable steps. It simplifies and prepares the equation for the next stages.

Isolation of Variable

The aim of isolating the variable is to have the variable by itself on one side of the equation. After distributing in our example, we have \( 22c - 8 = 36 \). The next step is to get \( 22c \) by itself. We do this by adding \( 8 \) to both sides to cancel out the \( -8 \). Thus:

• Add \( 8 \) to the left side: \( -8 + 8 = 0 \)
• Add \( 8 \) to the right side: \( 36 + 8 = 44 \)

Now, the equation reads \( 22c = 44 \). Keeping the variable isolated is critical because it sets the stage for the final step: solving for the variable itself by undoing any remaining multiplication or division.

Algebraic Manipulation

Algebraic manipulation is the ability to rearrange terms in an equation to make it easier to solve. This involves understanding and using properties of numbers to perform operations that will simplify the equation further. In our example, once \( 22c = 44 \) is achieved, the next step involves dividing both sides by \( 22 \) to solve for \( c \).

• Divide \( 22c \) by \( 22 \): \( \frac{22c}{22} = c \)
• Divide \( 44 \) by \( 22 \): \( \frac{44}{22} = 2 \)

Now, we've found \( c = 2 \)! By breaking down and manipulating the equation step-by-step, we ensure accuracy in finding the solution. This skill enhances problem-solving efficiency in math.