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Chapter 2: Problem 101

Simplify by combining like terms whenever possible. $$3(x+y)+4 x-y$$

Short Answer

Expert verified
7x + 2y

Step by step solution

01

Distribute the factor within the parentheses

Apply the distributive property to the expression: Distribute 3 through \(x+y\): \(3(x+y) = 3x + 3y\)
02

Rewrite the expression

Replace inside the original equation with the distributed terms: \(3x + 3y + 4x - y\)
03

Group like terms

Identify and group the like terms (those with the same variable): Terms with \(x\): \(3x + 4x\) Terms with \(y\): \(3y - y\)
04

Combine like terms

Add the coefficients of the grouped terms: \(3x + 4x = 7x\) \(3y - y = 2y\)
05

Write the simplified expression

Combine the results from step 4: \(7x + 2y\)

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Key Concepts

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

distributive property
Let's start with the basics of the distributive property. This property helps us break down expressions into simpler parts. It allows us to multiply a term outside the parenthesis by each term inside the parenthesis. For instance, in our exercise, we have to distribute 3 over the terms inside the parentheses. So, \[3(x + y)\] becomes \[3 \cdot x + 3 \cdot y = 3x + 3y\]. This step helps us get rid of the parentheses, making the expression easier to handle. Remember, distribution is straightforward multiplication applied to each term within the parentheses.
combining like terms
After distributing, your next job is to combine like terms. Like terms are terms that have the same variable raised to the same power. In our simplified expression \[3x + 3y + 4x - y\], we can spot like terms that need combining. The terms with x are \[3x\] and \[4x\], and the terms with y are \[3y\] and \[-y\]. To combine them, you add or subtract the coefficients. For example:
  • \[3x + 4x = 7x\]
  • \[3y - y = 2y\]
Combining like terms helps to simplify the expression further, reducing the number of different terms.
algebraic expressions
An algebraic expression is a combination of numbers, variables, and operation symbols. Let's break down the term step-by-step. A variable represents unknown values, like \(x\) or \(y\). Coefficients are the numbers in front of the variables, like the 3 in our expression. Operations can include addition, subtraction, multiplication, and division. Simplifying algebraic expressions involve applying the distributive property and combining like terms. By dealing with each part, step-by-step, we transform a complex expression into its simplest form. This gives us \[7x + 2y\], making it more understandable and manageable for solving algebra problems.

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Most popular questions from this chapter

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

Simplify by combining like terms whenever possible. $$0.085 x+28.75+0.072 x+54.8$$

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

Explain what is wrong with each of the following: (a) \(3 \cdot 2^{2} \stackrel{?}{=} 6^{2} \stackrel{?}{=} 36\) (b) \(3 x^{4} \stackrel{?}{=} 3 x \cdot 3 x \cdot 3 x \cdot 3 x\) (c) \(5^{4} \cdot 2 \stackrel{?}{=} 20 \cdot 2 \stackrel{?}{=}40\) (d) \(\left(3 x^{3}\right)^{2} \stackrel{?}{=} 6 x^{6}\) (e) \(-3^{4} \stackrel{?}{=}(-3) \cdot(-3) \cdot(-3)(-3) \stackrel{?}{=} 81\) (f) \(x^{4}+x^{5} \stackrel{?}{=} x^{9}\) (g) \(x^{2} \cdot x^{7} \stackrel{?}{=} x^{14}\) (h) \(\left(x^{2}\right)^{4} \stackrel{?}{=} x^{6}\)

Simplify by combining like terms whenever possible. $$x^{2}-2 x+x^{2}-x$$
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