Problem 47 - Intermediate Algebra

Question

Solve each equation. Check each result. See Example 5. $$ 3(k-4)=-36 $$

Step-by-Step Solution

Answer

The solution is \( k = -8 \).

Step 1: Distribute the 3

Begin by distributing the 3 on the left-hand side of the equation. This means multiplying the 3 by each term inside the parentheses: \( 3(k - 4) = 3k - 12 \). So, the equation becomes \( 3k - 12 = -36 \).

Step 2: Isolate the term with the variable

To isolate the term with the variable, add 12 to both sides of the equation to eliminate the -12: \( 3k - 12 + 12 = -36 + 12 \), resulting in \( 3k = -24 \).

Step 3: Solve for \( k \)

Divide both sides of the equation by 3 to solve for \( k \): \( \frac{3k}{3} = \frac{-24}{3} \). This simplifies to \( k = -8 \).

Step 4: Verify the solution

To check the result, substitute \( k = -8 \) back into the original equation to see if both sides are equal:1. Start with the original equation: \( 3(k-4) = -36 \).2. Replace \( k \) with \( -8 \): \( 3(-8 - 4) = -36 \).3. Simplify inside the parentheses: \( -8 - 4 = -12 \).4. Multiply: \( 3 \times (-12) = -36 \).Since both sides are equal, the solution \( k = -8 \) is correct.

Key Concepts

Distributive Property

The distributive property is a fundamental algebraic principle used to simplify expressions. It's particularly useful when dealing with terms inside parentheses. In our example, we started with the equation \[ 3(k-4) = -36. \] The distributive property allows us to remove the parentheses by multiplying each term inside by the number outside:

• Multiply 3 by \( k \) to get \( 3k \).
• Multiply 3 by \(-4\) to get \(-12 \).

This process transforms the equation into \[ 3k - 12 = -36, \] making it easier to solve. Using the distributive property is a helpful step in simplifying equations and setting the stage for isolating the variable.

Isolating Variables

After applying the distributive property, the next step is to isolate the variable. This means getting the variable on one side of the equation, by itself. In our example, we have:\[ 3k - 12 = -36. \]To isolate \( k \), we need to eliminate \(-12\) from the left side. We do this by adding 12 to both sides of the equation:
\[ 3k - 12 + 12 = -36 + 12, \]which simplifies to:\[ 3k = -24. \] This process involves balancing the equation by performing the same operation on both sides. Once isolated, dividing both sides by 3 to solve for \( k \) gives us:
\[ k = \frac{-24}{3} = -8. \]Isolating the variable is crucial, as it allows us to find the solution in a straightforward manner.

Checking Solutions

After finding a solution, it's vital to verify its correctness by substituting it back into the original equation. This step ensures that no errors were made. We start with our equation:\[ 3(k-4) = -36. \]Substituting \( k = -8 \) back into the equation, we have:
\[ 3(-8 - 4) = -36. \]First, simplify inside the parentheses:
\[ -8 - 4 = -12. \]Then, multiply:
\[ 3 \times (-12) = -36. \]Since the equation holds true, the solution \( k = -8 \) is verified as correct. Checking your solution is a simple yet effective strategy to confirm that you have correctly solved the equation. It provides reassurance and helps identify any missteps in the problem-solving process.