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Chapter 7: Problem 15

Simplify each expression. $$3(y-x)+5 y+8 x$$

Short Answer

Expert verified
The simplified expression of the problem is \(8y + 5x\).

Step by step solution

01

Distribute the '3'

Start by distributing the number preceding the brackets, which is '3', to every term inside the brackets. This gives: \(3y - 3x + 5y + 8x\)
02

Group Like Terms

Next, group like terms together. In other words, group the terms that are similar in nature. This involves grouping the 'y' terms and the 'x' terms separately. Giving: \( (3y + 5y) + (-3x + 8x)\)
03

Simplify Each Group

Simplify each group by performing the addition or subtraction as it stands. This provides: \(8y + 5x\)

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Key Concepts

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

Simplification of Expressions
Simplification of expressions is one of the foundational skills in algebra. When we simplify expressions, we aim to reduce them to their most compact and easily interpretable form. This process helps us work more efficiently with the expressions in equations and makes it easier to see relationships within mathematical problems.

When simplifying, take each term into account:
  • Distribute coefficients where applicable, as in the expression \(3(y-x)\) where the '3' multiplies each term inside the parentheses.
  • Combine like terms, which we'll explore further in another section, to condense the expression to its simplest form.
The end goal is to achieve a streamlined form that communicates all necessary information without excess arithmetic.
Distributive Property
The distributive property is a fundamental algebraic rule that allows us to distribute a single term over addition or subtraction inside parentheses. It enables transforming expressions like \(a(b+c)\) into \(ab + ac\). Essentially, you multiply the term outside the parentheses by each term within.

In the provided problem, we see this when the '3' outside the parentheses is applied to both \(y\) and \(-x\), resulting in \(3y - 3x\). This property ensures that each part of the expression is properly addressed and allows for simplification in subsequent steps.
  • Multiplying through removes the parentheses, making it easier to handle the expression.
  • It resolves complex nested expressions into simpler components.
Understanding and applying the distributive property is essential for mastering algebraic manipulations.
Combining Like Terms
Combining like terms is a crucial step in simplifying mathematical expressions. This involves grouping terms that have the same variables raised to the same power. Essentially, you are collecting and consolidating terms that are 'alike'.

In our example, after distributing, we have terms like \(3y\) and \(5y\) that can be combined. Likewise, the terms \(-3x\) and \(8x\) can be simplified into a single ""x"" term by addition:
  • Group 'y' terms (\(3y + 5y\)) together for a total of \(8y\).
  • Group 'x' terms (\(-3x + 8x\)) together for a total of \(5x\).
This method not only simplifies the expression but also makes solving equations easier by reducing the number of terms. Being proficient in combining like terms accelerates problem-solving in algebra.

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Most popular questions from this chapter

Graph the solution. $$4 x+3 y \leq 12$$

Graph the solution. $$3 x+2 y \geq 12$$

Graph the solution. $$\left\\{\begin{array}{l}\frac{x}{2}+\frac{y}{3} \geq 2 \\\\\frac{x}{2}-\frac{y}{2}<-1\end{array}\right.$$

Graph the solution. $$5 x+4 y \geq 20$$

Graph the solution set of each system of inequalities, when possible. SEE EXAMPLE 6. (OBJECTIVE 4) $$\left\\{\begin{array}{l} x+2 y \leq 3 \\ 2 x-y \geq 1 \end{array}\right.$$
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