Problem 20 - College Algebra with Modeling and Visualization

Question

Solve the equation and check your answer. $$ 6(5-3 t)=66 $$

Step-by-Step Solution

Answer

The solution is \( t = -2 \). After checking, it satisfies the original equation.

Step 1: Simplify the Equation

We start by distributing the 6 on the left-hand side of the equation. Multiply 6 by both terms inside the parentheses: \[ 6 imes 5 - 6 imes 3t = 66 \] which simplifies to: \[ 30 - 18t = 66 \]

Step 2: Isolate the Variable

Next, we move towards isolating the variable \( t \). Start by subtracting 30 from both sides of the equation: \[ 30 - 18t - 30 = 66 - 30 \] which simplifies to: \[ -18t = 36 \]

Step 3: Solve for the Variable

To solve for \( t \), divide both sides of the equation by -18: \[ \frac{-18t}{-18} = \frac{36}{-18} \] This gives us: \[ t = -2 \]

Step 4: Check the Solution

To verify the solution, substitute \( t = -2 \) back into the original equation: \[ 6(5 - 3(-2)) = 66 \] Simplify inside the parentheses: \[ 5 + 6 = 11 \] Then: \[ 6 \times 11 = 66 \] Since both sides equal 66, our solution \( t = -2 \) is correct.

Key Concepts

Distributive Property

In solving linear equations, the distributive property is a powerful tool. It allows us to simplify expressions and equations for easier handling. The distributive property states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately and then summing the results. In mathematical terms:
\[ a(b + c) = ab + ac \]
In the exercise, we applied this property to simplify the expression on the left side of the equation: \( 6(5 - 3t) \). By distributing the 6, we computed:

• \( 6 \times 5 = 30 \)
• \( 6 \times -3t = -18t \)

This leads to the simplified equation \( 30 - 18t = 66 \). Using the distributive property helps break down complex expressions into more manageable parts, making solving straightforward.

Isolating Variables

When solving equations, our goal is often to find the value of a variable, making isolating the variable a crucial step. This means rearranging the equation to get the variable on one side and everything else on the other. In our example, we needed to isolate \( t \) from the equation \( 30 - 18t = 66 \).
Here's how we did it:

• First, we subtracted 30 from both sides: \( 30 - 18t - 30 = 66 - 30 \), simplifying to \( -18t = 36 \).
• Next, we divided both sides by -18 to solve for \( t \): \( \frac{-18t}{-18} = \frac{36}{-18} \), resulting in \( t = -2 \).

These steps ensure the variable stands alone, helping us clearly see its value. Isolating allows us to focus on finding accurate solutions quickly.

Verifying Solutions

Verification is a crucial step in solving equations, as it confirms that the solution is correct. After finding the solution \( t = -2 \), it is important to check that it satisfies the original equation \( 6(5 - 3t) = 66 \).
To do this, you substitute \( t = -2 \) back into the equation:

• Replace \( t \) with -2: \( 6(5 - 3(-2)) \).
• Simplify inside the parentheses: \( 5 + 6 = 11 \).
• Then calculate \( 6 \times 11 = 66 \).

Since both sides equal 66, verifying that \( t = -2 \) is correct shows a reliable solution. Always remember, the verification step ensures your solution is not only well calculated but also functionally accurate.