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Chapter 1: Problem 10

Simplify the algebraic expressions by combining similar terms. $$ -x y+z-8 x y-7 z $$

Short Answer

Expert verified
The simplified expression is \(-9xy - 6z\).

Step by step solution

01

Identify Like Terms

In the expression \(-xy + z - 8xy - 7z\), identify the like terms. Like terms are terms that have the same variables raised to the same power. Here, \(-xy\) and \(-8xy\) are like terms as they both contain \(xy\), and \(z\) and \(-7z\) are like terms as they both contain \(z\).
02

Combine the 'xy' Terms

Add the coefficients of the \(xy\) terms: \(-1xy\) and \(-8xy\) can be combined to give \((-1) + (-8) = -9\). Thus, \(-xy - 8xy = -9xy\).
03

Combine the 'z' Terms

Add the coefficients of the \(z\) terms: \(1z\) and \(-7z\) can be combined to give \(1 + (-7) = -6\). Thus, \(z - 7z = -6z\).
04

Write the Simplified Expression

Combine the results from Steps 2 and 3 to write the fully simplified expression. Substituting the combined terms gives:\(-9xy - 6z\).

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Key Concepts

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

Like Terms
When simplifying algebraic expressions, understanding the concept of like terms is essential. Like terms are parts of an expression that contain the same variables, with each variable raised to the same power. This means that the structure of the term in terms of variable composition and exponents is identical.
For example:
  • In the expression \(-xy + z - 8xy - 7z\), the terms \(-xy\) and \(-8xy\) are like terms because they both include the variable combination \(xy\).
  • Similarly, \(z\) and \(-7z\) are like terms as both involve the variable \(z\) raised to the power of one (or simply \(z\)).
Identifying like terms is the first step towards simplifying expressions. This process helps recognize parts of the expression that can be combined, leading to a simpler form.
Combine Coefficients
Once like terms are identified, the next step is to combine them. This involves calculating the sum or difference of their coefficients. The coefficient is the numerical part that multiplies the variable(s) within a term.
For instance:
  • Consider the terms \(-xy\) and \(-8xy\). Here, the coefficients are \(-1\) and \(-8\) respectively. Combining them gives us \((-1) + (-8) = -9\), resulting in the simplified form \(-9xy\).
  • Similarly, with the terms \(z\) and \(-7z\), the coefficients are \(1\) and \(-7\). Adding these results in \(1 + (-7) = -6\), simplifying the sum to \(-6z\).
By focusing on the coefficients and summing them up, we make the expression leaner and easier to manage.
Algebraic Expressions
An algebraic expression is a mathematical phrase involving numbers, variables, and operators (such as addition or subtraction). Simplifying such expressions is crucial in making them more straightforward and easier to work with. This process typically involves combining like terms and calculating sums or differences in the coefficients.
In our example, the expression \(-xy + z - 8xy - 7z\) can be transformed through finding like terms and combining their coefficients:
  • First, we deal with like terms such as \(-xy\) and \(-8xy\), as well as \(z\) and \(-7z\).
  • Secondly, by applying the principles of combining coefficients, we arrive at a simplified expression \(-9xy - 6z\).
This demonstrates how recognizing structures and patterns in algebraic expressions leads to simplicity, making equations much more manageable in various applications.

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Most popular questions from this chapter

Evaluate the algebraic expressions for the given values of the variables. $$ 2(x-1)-(x+2)-3(2 x-1), \quad x=-1 $$

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Translate each English phrase into an algebraic expression and use \(n\) to represent the unknown number. The quotient of 50 and a number

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