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Chapter 0: Problem 86

Simplify each expression. $$ -(10 x-z)-2(8 x-4 z)-3 x $$

Short Answer

Expert verified
-29x + 9z

Step by step solution

01

Distribute the Negative Sign

The first step is to distribute the negative sign across the terms inside the parentheses. This gives us: \[ -(10x - z) = -10x + z \]So the expression becomes:\[ -10x + z - 2(8x - 4z) - 3x \]
02

Distribute the -2

Next, distribute the -2 across the terms inside the second set of parentheses:\[ -2(8x - 4z) = -2 \times 8x + -2 \times -4z = -16x + 8z \]So the expression now becomes:\[ -10x + z - 16x + 8z - 3x \]
03

Combine Like Terms

Combine the like terms by adding or subtracting their coefficients. The terms involving x are:\[ -10x - 16x - 3x = -29x \]The terms involving z are:\[ z + 8z = 9z \]Thus, the simplified expression is:\[ -29x + 9z \]

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Key Concepts

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

Distribute Negative Sign
When you see a negative sign outside of parentheses, it means you need to distribute it to each term inside. This process changes the sign of each term. In our example, we start with:
  • o - (10x - z)
We change the signs within the parentheses:
  • - 10x + z
This modifies our expression to:
  • -10x + z - 2(8x - 4z) - 3x
Always distribute the negative sign first to avoid mistakes later on.

Combine Like Terms
Combining like terms is crucial for simplifying expressions. Like terms are terms that have the same variable raised to the same power. For instance, -10x, -16x, and -3x are like terms because they all contain the variable x raised to the same power of one.
The steps to combining like terms here are: First, identify all the like terms for each variable and group them. Then, add or subtract their coefficients. For our variables involving 'x':
  • -10x - 16x - 3x = -29x
For our 'z' terms:
  • z + 8z = 9z

Combining like terms makes the expression simpler and more organized.

Coefficients in Algebra
Coefficients are the numerical part of terms involving variables. They provide scale to the variables' values.
For example, in the term -10x, -10 is the coefficient and x is the variable. Similarly, in 8z, the number 8 is the coefficient and z is the variable.
In the context of our simplification problem, after distributing the negative sign and combining like terms, the coefficients play a key role. We add and subtract these coefficients to simplify the expression.
  • -10x, -16x, and -3x combined give us -29x
  • z and 8z combined give us 9z
Understanding coefficients helps you manage and simplify algebraic expressions efficiently.

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