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Chapter 1: Problem 36

Simplify each algebraic expression by combining similar terms. $$14 y-17 y-19 y$$

Short Answer

Expert verified
The simplified expression is \(-22y\).

Step by step solution

01

Identify Like Terms

Identify the like terms within the algebraic expression. Like terms are terms that have the same variable raised to the same power. In the expression \(14y - 17y - 19y\), all three terms \(14y\), \(-17y\), and \(-19y\) are like terms because they all contain the variable \(y\) raised to the first power.
02

Combine Like Terms

Once you have identified the like terms, you combine them by adding or subtracting their coefficients. For the expression \(14y - 17y - 19y\), add the coefficients:\[14 + (-17) + (-19)\]Calculate the sum:\[14 - 17 - 19 = -22\]Thus, the simplified expression is \(-22y\).

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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, like terms are terms that have the exact same variable with the same exponent. This means they have the same letter or letters and the same power. These terms can be combined through basic operations like addition and subtraction. When looking at the expression \(14y - 17y - 19y\), we notice that all terms include the variable \(y\) raised to the first power, making them like terms. Recognizing like terms is crucial because it helps simplify expressions more efficiently.
The Process of Combining Like Terms
Combining like terms is an essential step in simplifying algebraic expressions. This process involves adding or subtracting the coefficients of like terms. Here's how you can do it effectively:
  • Identify all like terms in the expression. In our example \(14y - 17y - 19y\), each term is a like term because they share the variable \(y\).
  • Add or subtract the coefficients, which are the numbers in front of the variables. For these terms, the coefficients are 14, -17, and -19.
  • When you compute \(14 + (-17) + (-19)\), the result is \(-22\).
By combining like terms, you can transform complicated expressions into simpler forms, like we simplified to \(-22y\). This makes working with such expressions much more manageable.
Breaking Down Coefficients
Coefficients are the numerical part of terms that are multiplied by the variable. In other words, they are the numbers that come before the variables in an algebraic expression. In the expression \(14y - 17y - 19y\), the coefficients are 14, -17, and -19. Here's how to interpret coefficients effectively:
  • The coefficient tells you how many times the variable is counted. For example, 14 in \(14y\) means 14 instances of \(y\).
  • A negative sign before a coefficient, like -17 or -19, indicates subtracting the corresponding instances of the variable.
Understanding coefficients is fundamental to properly combining like terms, as they define the operations performed on the variables. When combined correctly, they simplify the expression as we've done to get \(-22y\).

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Most popular questions from this chapter

Simplify each numerical expression. Don't forget to take advantage of the properties if they can be used to simplify the computation. $$-21+22-23+27+21-19$$

Find the value of \(\frac{h\left(b_{1}+b_{2}\right)}{2}\) for each set of values for the variables \(h, b_{1}\), and \(b_{2}\). (Subscripts are used to indicate that \(b_{1}\) and \(b_{2}\) are different variables.) \(h=9, b_{1}=12\), and \(b_{2}=16\)

Simplify each algebraic expression by combining similar terms. $$14 x y-7-19 x y-6$$

Simplify each numerical expression. Don't forget to take advantage of the properties if they can be used to simplify the computation. $$-9(14)-7(-16)$$

Simplify each algebraic expression and then evaluate the resulting expression for the given values of the variables. \((x-6)-(x+12)\) for \(x=-3\)
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