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Chapter 1: Problem 112

Simplify each expression. $$ 6.5 y-4.4(1.8 x-3.3)+10.95 $$

Short Answer

Expert verified
The simplified expression is \(6.5y - 7.92x + 25.47\).

Step by step solution

01

Distribute the Negative Coefficient

First, distribute \(-4.4\) into the expression \(1.8x - 3.3\) inside the parentheses. Distributing means multiplying each term inside by \(-4.4\).\[-4.4 imes 1.8x + (-4.4) imes (-3.3) = -7.92x + 14.52\]
02

Substitute and Simplify

Substitute the result of the distribution back into the expression. Replace \(-4.4(1.8x - 3.3)\) with \(-7.92x + 14.52\):\[6.5y - 7.92x + 14.52 + 10.95\]Now, combine the constants \(14.52\) and \(10.95\).
03

Combine Like Terms

Add the constant terms \(14.52\) and \(10.95\) together:\[14.52 + 10.95 = 25.47\]Substitute back:\[6.5y - 7.92x + 25.47\]
04

Final Simplified Expression

The expression is simplified to:\[6.5y - 7.92x + 25.47\]

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Key Concepts

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

Distributive Property
The distributive property is a fundamental principle used to simplify algebraic expressions. It allows us to "distribute" a multiplier across terms within parentheses. For instance, in the expression \(-4.4(1.8x - 3.3)\), we need to distribute \(-4.4\) to both \(1.8x\) and \(-3.3\). This means:
  • Multiplying \(-4.4\) by \(1.8x\) resulting in \(-7.92x\).
  • Multiplying \(-4.4\) by \(-3.3\) resulting in \(14.52\).
Thus, the distributed result is \(-7.92x + 14.52\). Simplifying expressions through distribution helps in breaking down complex equations into manageable parts. Remember:
  • Always distribute to all terms inside the parentheses.
  • Be careful with signs; distributing a negative number changes the sign of each term inside the parentheses.
Combining Like Terms
Combining like terms is an essential step in simplifying algebraic expressions. Like terms are terms that have the same variable raised to the same power. For example, in the expression we've simplified, \(6.5y - 7.92x + 14.52 + 10.95\), the like terms are the constants: \(14.52\) and \(10.95\).
To combine these constants, simply add them together:
  • \(14.52 + 10.95 = 25.47\)
Substituting this result back into the expression, we get \(6.5y - 7.92x + 25.47\).
  • It's important to remember that you only combine terms that have the same variable parts.
  • When combining terms, think of them as fruit: you can combine apples with apples but not apples with oranges!
Negative Coefficients
Negative coefficients appear when a negative number multiplies a variable in an algebraic expression. They can affect the signs of terms during multiplication and distribution.
In our example, distributing the negative coefficient \(-4.4\) changed the sign of each term inside the parentheses:
  • \(1.8x\) became \(-7.92x\).
  • \(-3.3\) became \(14.52\).
Negative coefficients signal that the direction (or sign) of the multiplication will change. Managing these correctly is crucial:
  • Remember that multiplying two negative numbers results in a positive product.
  • Be careful during distribution to ensure all signs are correctly applied.
Always double-check operations involving negative coefficients to avoid mistakes in the sign of the terms.

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Most popular questions from this chapter

Simplify each expression. See Examples I through 11 . $$ \frac{\frac{1}{3} \cdot 9-7}{3+\frac{1}{2} \cdot 4} $$

Choose the fraction (s) equivalent to the given fraction. (There may be more than one correct choice.) \(\frac{-x}{y} \quad\) a. \(\frac{x}{-y}\) b. \(-\frac{x}{y} \quad\) c. \(\frac{x}{y}\) d. \(\frac{-x}{-y}\)

Simplify each expression. See Examples \(11,14,\) and \(15 .\) $$ \frac{3}{4} b-\frac{1}{2}+\frac{1}{6} b-\frac{2}{3} $$

Use the distributive property to find the product. See Example II. $$ 0.5 x(6 y-3) $$

Simplify each expression. See Examples I through 11 . $$ 3\\{-2+5[1-2(-2+5)]\\} $$
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