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Chapter 1: Problem 80

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

Short Answer

Expert verified
43.2t - 28.8

Step by step solution

01

- Distribute the Constants

Distribute the constant outside each parenthesis to each term inside the parenthesis.\[8.4(6t - 6) + 2.4(9 - 3t) = 8.4 \times 6t - 8.4 \times 6 + 2.4 \times 9 - 2.4 \times 3t\]\[= 50.4t - 50.4 + 21.6 - 7.2t\]
02

- Combine Like Terms

Combine the terms involving \(t\) and the constant terms separately.\[50.4t - 7.2t - 50.4 + 21.6\]Combine the coefficients of \(t\):\[50.4t - 7.2t = 43.2t\]Combine the constant terms:\[-50.4 + 21.6 = -28.8\]
03

- Write the Simplified Expression

Combine the simplified terms from Step 2 to form the final expression:\[43.2t - 28.8\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

The Distributive Property
The Distributive Property is a fundamental concept in algebra. It's used to eliminate parentheses by distributing a factor across terms inside parentheses. For example, in the expression \(8.4(6t - 6) + 2.4(9 - 3t)\), the first step is to multiply each term inside the parentheses by the factor outside. This simplifies to \(8.4 \times 6t - 8.4 \times 6 + 2.4 \times 9 - 2.4 \times 3t\).
Distributing helps to make the expression easier to work with. Remember, each term inside the parentheses must be multiplied by the constant outside them. This calls for basic multiplication of each coefficient with the numbers inside the parentheses.
Combining Like Terms
After using the Distributive Property, the next step in simplifying an expression is to combine like terms. Like terms are terms that contain the same variables raised to the same power. For example, \(50.4t - 7.2t\) are like terms because they both contain the variable \(t\). Combining these gives us \(43.2t\).
Similarly, \(-50.4 + 21.6\) are constants (terms without variables) and should be combined as well. This simplifies to \(-28.8\).
When combining like terms, simply add or subtract their coefficients. This reduces the expression to its simplest form, making it easier to understand and use in further calculations.
Algebraic Simplification
The final step in our example is putting it all together into a simplified form. Algebraic simplification means to rewrite the expression in its most reduced form without changing its value.
From our previous steps, we've simplified the original expression \(8.4(6 t-6)+2.4(9-3 t)\) to \(43.2t - 28.8\). This final form is much simpler and more manageable.
Simplifying expressions helps to solve equations more efficiently and makes it easier to understand the relationships between variables. Always remember the key steps: distribute, combine like terms, and simplify.

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Most popular questions from this chapter

Decide whether each statement is an example of the commutative, associative, identity, inverse, or distributive property. See Examples \(1,2,3,5,6,7,\) and \(9 .\) $$2(x+y)=2 x+2 y$$

Simplify each expression. $$ 6-3 z-2 z-5+z-3 z $$

Translate each phrase into a mathematical expression. Use \(x\) as the variable. Combine like terms when possible. A number multiplied by \(5,\) subtracted from the sum of 14 and eight times the number

Write a fraction with denominator 24 that is equivalent to \(\frac{5}{8}\)

To find the average (mean) of numbers, we add the numbers and then divide the sum by the number of terms added. For example, to find the average of \(14,8,3,9,\) and \(1,\) we add them and then divide by 5. $$ \frac{14+8+3+9+1}{5}=\frac{35}{5}=7 \leftarrow \text { Average } $$ Find the average of each group of numbers. All even integers between \(-18\) and \(4,\) inclusive
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