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Chapter 1: Problem 8

Simplify the given algebraic expressions. $$-4 C+L-6 C$$

Short Answer

Expert verified
The expression simplifies to \(-10C + L\).

Step by step solution

01

Identify Like Terms

In the expression \(-4C + L - 6C\), the terms \(-4C\) and \(-6C\) are like terms because they both contain the variable \(C\). The term \(L\) is not like any other term in this expression and should be kept separate.
02

Combine Like Terms

Combine the like terms \(-4C\) and \(-6C\). This is done by adding the coefficients of these terms:\(-4 + (-6) = -10\).Thus, the expression simplifies to \(-10C + L\).
03

Conclusion

The simplified form of the expression \(-4C + L - 6C\) is \(-10C + L\).

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Key Concepts

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

Understanding Like Terms
In algebra, terms are called "like terms" when they have exactly the same variable components raised to the same powers. For example, in the expression
  • \(-4C + L - 6C\),
the terms \(-4C\) and \(-6C\) are like terms because they both contain the variable \(C\) to the first power. Identifying like terms is crucial because it lets you know which parts of an expression can be combined.
A few key points to remember when identifying like terms include:
  • Constant numbers (those without variables) are like terms with each other, such as in \(3\) and \(5\).
  • If the variable is different or the powers of the variables do not match, they are not like terms. For example, \(3x^2\) and \(3x\) are not like terms.
  • Different variables, like \(x\) and \(y\), also cannot be combined into like terms.
Understanding this sets the foundation for combining terms effectively in algebraic expressions.
Combining Like Terms
Once you identify like terms, the next step is to combine them. This process simplifies the expression and reveals a more straightforward version. Combining like terms involves the following steps:
  • Add or subtract the coefficients of the like terms. The coefficients are the numbers in front of the variables. In the expression \(-4C + L - 6C\), the like terms \(-4C\) and \(-6C\) have coefficients \(-4\) and \(-6\) respectively.
  • Perform the arithmetic operation on the coefficients. Here, you would add \(-4\) and \(-6\) to get \(-10\).
  • Keep the variable and its power unchanged. This means \(-4C + (-6C)\) becomes \(-10C\).
Always remember that only like terms can be combined. After combining, you will typically have fewer terms, leading to a simpler, more manageable expression.
Simplifying Expressions
Simplifying expressions is all about making the expression as clear and brief as possible. It involves:
  • Identifying and combining like terms as discussed previously.
  • Clearing up any unnecessary complexity while maintaining the expression's original value.
For the expression \(-4C + L - 6C\), after combining the like terms, you're left with \(-10C + L\). Here, the expression has been brought down to its simplest form because no further like terms exist to combine.
Simplification doesn't just make expressions easier to read; it also makes them easier to work with, especially when solving equations or performing further operations. It often marks the initial and essential step in problem-solving within algebra.

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