Problem 88 - Introductory Algebra for College Students

Question

Simplify each algebraic expression. $$-y+9 y$$

Step-by-Step Solution

Answer

The simplified expression is 8y.

Step 1: Identify Like Terms

In the expression -y+9y, both -y and 9y are like terms because they both contain the variable 'y'.

Step 2: Combine Like Terms

We combined like terms by adding their coefficients together. So, -1 + 9 = 8. Therefore, -y+9y simplifies to 8y.

Key Concepts

Like Terms in Algebra

Understanding like terms is essential for simplifying algebraic expressions effectively. Like terms are terms in an algebraic expression that have the same variable raised to the same power. Essentially, they are terms that you can combine. For example, in the expression \(2x + 3x\), both terms are like terms since they both contain the variable \(x\) with the same exponent 1 (implied). However, \(2x\) and \(2x^2\) are not like terms because the variables are raised to different powers. Similarly, \(3x\) and \(3y\) do not qualify as like terms because they involve different variables.

When simplifying an algebraic expression, one of the first steps is identifying like terms. This is because only like terms can be combined to simplify the expression. In our exercise, \( -y + 9y \), both terms are indeed like terms because the variable \(y\) has no exponent, which means it's raised to the power of 1, and it is identical in both terms.

Importance of Identifying Like Terms

Proper identification of like terms is critical to carrying out operations like addition and subtraction in algebra. Without pairing like terms correctly, you may end up with an incorrect simplification of the expression. Identifying like terms often leads to a more consolidated and manageable expression, making it easier to work with and understand.

Combining Like Terms

Once like terms are identified, the next step is combining them to simplify the expression. Combining like terms involves adding or subtracting their coefficients — the numerical factors that multiply the variables. This is an application of the distributive property of multiplication over addition and subtraction.

For instance, in our given exercise \( -y + 9y\), the like terms with the variable \(y\) can be combined by adding the coefficients \( -1 \) and \( 9 \) together. It's important to note that the negative sign in front of \(y\) denotes a coefficient of \( -1 \). Therefore, we have \( -1 + 9 = 8\), and the combined term is \(8y\).

Steps to Combine Like Terms

• Identify the like terms in the expression.
• Align their coefficients.
• Perform the required addition or subtraction.
• Write the simplified expression with the combined like terms.

Combining like terms not only simplifies the expression but also lays the foundation for solving equations, which often involves isolating variables.

Algebraic Expression Coefficients

Coefficients in algebraic expressions are the numerical parts of the terms that are multiplied by the variables. They provide a multiplicative factor to the variable, influencing the term's value. In the expression \( -y + 9y\), \( -1 \) and \( 9 \) are the coefficients of \(y\). It's crucial to recognize that every variable term has a coefficient, even if it's not explicitly shown. For example, \(x\) is understood to have a coefficient of \(1\), while \( -x\) has a coefficient of \( -1\).

In simplifying algebraic expressions, coefficients play a key role. They determine the magnitude and direction (positive or negative) that the variables contribute to the overall expression.

Handling Negative Coefficients

Dealing with negative coefficients requires special attention to signs during simplification. As demonstrated in our exercise, the negative coefficient \( -1 \) in \( -y\) is subtracted from the coefficient \( 9 \) in \( 9y\). The operation of combining coefficients should respect the signs to accurately represent the sum or difference of the quantities involved.