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Chapter 1: Problem 2

Perform the indicated operation and simplify. $$(4 y-5)-(10 y-7)$$

Short Answer

Expert verified
-6y + 2

Step by step solution

01

- Distribute the negative sign

The expression given is \( (4y - 5) - (10y - 7) \). To distribute the negative sign, rewrite the expression as \( (4y - 5) + (-1)(10y - 7) \). Then, distribute -1 through the second parentheses: \( (4y - 5) + (-10y + 7) \).
02

- Combine like terms

Combine the terms with \ y \ and the constant terms separately. Add the \ y \ terms: \( 4y + (-10y) = -6y \). Add the constant terms: \( -5 + 7 = 2 \). The simplified expression is \ -6y + 2 \.

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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 an important concept in algebra that allows us to simplify expressions where a term is multiplied by terms inside parentheses. This property states: \[ a(b + c) = ab + ac \].
In simpler terms, you the term outside the parenthesis is multiplied by each term inside the parenthesis.
Let's look at our example expression \((4y - 5) - (10y - 7)\). The second part of the expression is \((10y - 7)\) which is subtracted. To make it easier to simplify, we distribute the negative sign:\((10y - 7)\) becomes \((-1 \times 10y) + (-1 \times -7)\), which simplifies to \(-10y + 7\).This gives us \((4y - 5) + (-10y + 7)\).
combining like terms
Combining like terms is a crucial step in simplifying algebraic expressions. It means adding or subtracting terms that have the same variable part.
For example, in the expression \((4y - 5) + (-10y + 7)\), we need to combine the \(y\) terms and the constant terms separately:
  • Combine \(4y\) with \(-10y\): \4y + (-10y) = -6y\
  • Combine the constants \(-5\) and \7\): \-5 + 7 = 2\.
So the simplified expression after combining like terms is \(-6y + 2)\.
negative sign distribution
Negative sign distribution involves distributing a negative sign (or \(-1\)) across an entire expression inside parentheses. This step is often necessary to handle expressions with subtraction.
Let's consider our original example: \((4y - 5) - (10y - 7)\). Initially, we rewrite it by distributing the negative sign as follows:
  • Rewrite the expression with \(-1\): \((4y - 5) + (-1)(10y - 7)\).
  • Distribute \(-1\) through the second parentheses: \(-1 \times 10y + -1 \times -7\).
  • This simplifies to \(-10y + 7\).
So, our expression now looks like \((4y - 5) + (-10y + 7)\), making it easier to combine like terms.

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Most popular questions from this chapter

Determine the set of values for \(x\) for which the radical expression would produce a real number. For example, the expression \(\sqrt{x-1}\) is a real number if \(x-1 \geq 0\) or equivalently, \(x \geq 1\). a. \(\sqrt{x+7}\) b. \(\sqrt[3]{x+7}\)

Martha is in a boat in the ocean \(48 \mathrm{mi}\) from point \(A\), the closest point along a straight shoreline. She needs to dock the boat at a marina \(x\) miles farther up the coast, and then drive along the coast to point \(B, 96 \mathrm{mi}\) from point \(A\). Her boat travels \(20 \mathrm{mph}\), and she drives \(60 \mathrm{mph}\). If the total trip took \(4 \mathrm{hr}\), determine the distance \(x\) along the shoreline.

a. Write an absolute value equation or inequality to represent each statement. b. Solve the equation or inequality. Write the solution set to the inequalities in interval notation. The distance between a number \(x\) and 3 on the number line is 8 .

Write an absolute value inequality that represents the statement. \(7-y<-1\) or \(7-y>1\)

Solve the equations. \(-3=-|c-7|+1\)
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