Problem 104 - Elementary and Intermediate Algebra

Question

Simplify each expression, if possible. $$ -6(3 t-6)-3(11 t-3) $$

Step-by-Step Solution

Answer

The simplified expression is \(-51t + 45\).

Step 1: Distribute the Constants

First, distribute each constant outside the parentheses to each term inside the parentheses. Start with \(-6(3t - 6)\), which becomes \(-18t + 36\). Now do the same for \(-3(11t - 3)\), and you get \(-33t + 9\). So the expression becomes \(-18t + 36 - 33t + 9\).

Step 2: Combine Like Terms

Now that the expression is simplified to \(-18t + 36 - 33t + 9\), identify and combine the like terms. The terms involving \(t\) are \(-18t\) and \(-33t\). Adding these coefficients, we get \(-51t\). The constant terms, \(36\) and \(9\), combine to make \(45\). So the expression simplifies to \(-51t + 45\).

Key Concepts

Distribution Property

The distribution property is a fundamental algebraic tool that helps simplify expressions and solve equations. It allows you to multiply a single term by each term within a parenthesis, effectively "distributing" it across all the terms inside.
When you encounter an expression like \[ -6(3t - 6) - 3(11t - 3) \]this property helps you manage and simplify it by multiplying each term inside the parentheses with the term outside.
Start by distributing \( -6 \) across \( 3t - 6 \):

• Multiply \( -6 \) by \( 3t \) to get \( -18t \)
• Multiply \( -6 \) by \( -6 \) to get \( 36 \)

This results in the expression \( -18t + 36 \).
Next, distribute \( -3 \) across \( 11t - 3 \):

• Multiply \( -3 \) by \( 11t \) to get \( -33t \)
• Multiply \( -3 \) by \( -3 \) to get \( 9 \)

This gives you \( -33t + 9 \).
Using the distribution property ensures that every term is accounted for and helps to clear the parentheses, setting the stage for further simplification.

Combining Like Terms

After using the distribution property, the next step is combining like terms. This process involves identifying terms with the same variable and the same power and then summing or subtracting their coefficients.
In the expression \( -18t + 36 - 33t + 9 \), you notice that terms involving \( t \) can be paired together:

• \( -18t \) and \( -33t \) are like terms because they both involve the variable \( t \).

By adding these terms (\( -18t - 33t \)), the result is \( -51t \).
Similarly, constant terms, which are numbers without variables, can also be combined:

• \( 36 \) and \( 9 \) are like because neither has a variable component.

Adding these together yields \( 45 \).
The process of combining like terms helps condense the expression, making it easier to interpret and solve.

Simplification

The overall goal of simplification is to reduce the complexity of an algebraic expression, making it as concise and straightforward as possible. After distributing and combining like terms, simplification often reveals a cleaner expression.
Consider the expression that arises from our earlier steps: \( -51t + 45 \).
At this point, simplification means double-checking your work to ensure there are no more like terms to combine.

• The terms \( -51t \) and \( 45 \) cannot be simplified further together since one involves a variable and the other is a constant.

Thus, \( -51t + 45 \) is the simplest form of the original algebraic expression.
Simplification enables you to see the essential parts of an expression, providing clarity and ease of use for further mathematical operations or solving equations.