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Chapter 2: Problem 5

Simplify each expression as completely as possible. $$-2(x-3 y)+5(y-x)$$

Short Answer

Expert verified
\(-7x + 11y\)

Step by step solution

01

Distribute the Constants

Distribute the \( -2 \) into the first parenthesis and the \( 5 \) into the second parenthesis:
02

Apply Distribution

Apply the distribution to each term inside the parentheses: \( -2(x-3y) = -2x + 6y \) and \( 5(y-x) = 5y - 5x \).
03

Combine the Like Terms

Combine the like terms from both expressions: \( -2x + 6y + 5y - 5x = -7x + 11y \)

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Key Concepts

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

Distribution
The distribution in algebra involves spreading a number outside the parentheses to each term inside the parentheses.

In our exercise, we have the expression \( -2(x-3 y)+5(y-x) \). To simplify it, we distribute \-2 \ into \(x-3y\) and 5 into \(y-x\).

Distribution works by multiplying the external number with each term inside the parentheses separately.
When we apply this step:
  • \( -2 \cdot x -2 \cdot (-3y) \) gives \(-2x + 6y\).
  • \( 5 \cdot y - 5 \cdot x \) gives \(5y - 5x\).

Understanding distribution is key to manipulating algebraic expressions as it helps in breaking down complex expressions into simpler parts.
Combining Like Terms
Combining like terms is essential for simplifying algebraic expressions.

Like terms are the terms in an expression that have the same variables raised to the same power. For instance, in our simplified parts: \( -2x + 6y + 5y - 5x \), we can identify like terms.

These involve terms with \(x\) and terms with \(y\).
  • Group the \(x\)-terms: \(-2x - 5x \)
  • Group the \(y\)-terms: \( 6y + 5y \)

Adding these like terms together:
  • \(-2x - 5x = -7x \)
  • \((6y + 5y = 11y \)

Our final expression is simplified to: \(-7x + 11y\). This makes it easier to understand and solve equations.
Algebraic Manipulation
Algebraic manipulation refers to the process of rearranging and combining expressions to make them simpler. It involves using distribution and combining like terms effectively.

Our example \( -2(x-3 y)+5(y-x) \) was addressed step-by-step:
  • First, using distribution to unpack the parentheses.
  • Then, recognizing and combining like terms.

These techniques transform a complex expression into a more manageable one.

Here's a quick recap:

Step 1: Distribute the constants:
\( -2(x-3 y) \rightarrow -2x + 6y \) and \( 5(y-x) \rightarrow 5y - 5x \)

Step 2: Combine like terms:
\(-2x - 5x = -7x \) and \((6y + 5y = 11y \)

The final simplified expression is \(-7x + 11y\). Mastering these skills will make you proficient in algebra.

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Most popular questions from this chapter

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$(y-2 z)(y+3 z)$$

Simplify by combining like terms whenever possible. $$7 y^{2}(y+2)+2 y\left(5 y^{2}\right)$$

Simplify by combining like terms whenever possible. $$5 x\left(x^{2}+3\right)+2 x\left(3+x^{2}\right)$$

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$-x^{2}(-z)$$

Evaluate the given expression for \(x=2, y=-3,\) and \(z=-4.\) $$(z-3 x)^{2}$$
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