Problem 92 - Beginning and Intermediate Algebra

Question

Simplify each expression. \(8.4(6 t-6)+2.4(9-3 t)\)

Step-by-Step Solution

Answer

The simplified expression is \(43.2t - 28.8\).

Step 1 - Distribute the Constants

Apply the distributive property to both parts of the expression. For the first part, distribute 8.4 to both terms inside the parentheses: \[8.4 \times 6t - 8.4 \times 6\] For the second part, distribute 2.4 to both terms inside the parentheses: \[2.4 \times 9 - 2.4 \times 3t\]

Step 2 - Perform the Multiplication

Calculate the results of the multiplication from the distribution: \[8.4 \times 6t = 50.4t\] \[8.4 \times 6 = 50.4\]\[2.4 \times 9 = 21.6\] \[2.4 \times 3t = 7.2t\] Place these results back into the expression: \[50.4t - 50.4 + 21.6 - 7.2t\]

Step 3 - Combine Like Terms

Group and combine the like terms from the simplified expression:\[50.4t - 7.2t - 50.4 + 21.6\]Combine the terms with 't': \[50.4t - 7.2t = 43.2t\]Combine the constant terms:\[-50.4 + 21.6 = -28.8\]The simplified expression is: \[43.2t - 28.8\]

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra that allows us to simplify expressions. It states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. In mathematical terms, it's represented as:
\(a(b + c) = ab + ac\).
In the exercise, we apply the distributive property to both parts of the expression. For the first part, we distribute 8.4 to both terms inside the parentheses:
\(8.4 \times 6t - 8.4 \times 6\).
For the second part, we distribute 2.4 to both terms inside the parentheses:
\(2.4 \times 9 - 2.4 \times 3t\).
This process ensures that each term within the parentheses is multiplied by the constant outside, simplifying the expression further for the next steps.

Combining Like Terms

Combining like terms is another essential skill in algebra. Like terms are terms that have the same variable raised to the same power. These terms can be added or subtracted together to simplify the expression.
In our example, after distributing the constants, we obtain the expression:
\(50.4t - 50.4 + 21.6 - 7.2t\).
We then group the like terms together:
\(50.4t - 7.2t\) and \(-50.4 + 21.6\).
Combining the terms with 't', we get:
\(50.4t - 7.2t = 43.2t\).
Combining the constant terms, we get:
\(-50.4 + 21.6 = -28.8\).
So, combining like terms helps us to simplify the expression to:
\(43.2t - 28.8\).

Multiplication of Constants

Multiplication of constants is a basic arithmetic operation but crucial in simplifying algebraic expressions. When multiplying constants by variables or other constants, each multiplication step needs to be computed accurately.
For instance, in our exercise, we perform the following multiplications:

• \(8.4 \times 6t = 50.4t\)
• \(8.4 \times 6 = 50.4\)
• \(2.4 \times 9 = 21.6\)
• \(2.4 \times 3t = 7.2t\)

These operations are straightforward but essential to get correct to simplify the expression accurately. Placing the results back into the expression, we get:
\(50.4t - 50.4 + 21.6 - 7.2t\). Properly handling the multiplication of constants ensures that our simplification steps are precise and help us reach the final simplified form.