Problem 88 - Algebra A Combined Function

Question

Simplify each expression by performing the indicated operation. Explain how you arrived at each answer. a. \(2 y+y\) b. \(2 y \cdot y\) c. \(-2 y-y\) d. \((-2 y)(-y)\)

Step-by-Step Solution

Answer

a. \(3y\), b. \(2y^2\), c. \(-3y\), d. \(2y^2\)

Step 1: Simplify by Combining Like Terms

In the expression \(2y + y\), both terms are 'like terms', as they both contain the variable \(y\). Combine them by adding the coefficients: \(2 + 1 = 3\). Thus, \(2y + y = 3y\).

Step 2: Multiply the Terms

In the expression \(2y \cdot y\), you multiply the coefficients and add the exponents of \(y\). Here, \(2 \cdot 1 = 2\) and for the exponents, \(1 + 1 = 2\). Hence, \(2y \cdot y = 2y^2\).

Step 3: Simplify by Subtracting Like Terms

In the expression \(-2y - y\), both terms are like terms with the variable \(y\). Combine them by adding the coefficients: \(-2 - 1 = -3\). So \(-2y - y = -3y\).

Step 4: Multiply Using the Negative Sign Rule

For the expression \((-2y)(-y)\), multiply the coefficients, taking into account that multiplying two negative numbers gives a positive result: \((-2) \times (-1) = 2\). Also, add the exponents of \(y\): \(1 + 1 = 2\). Hence, \((-2y)(-y) = 2y^2\).

Key Concepts

Understanding Like Terms

When simplifying algebraic expressions, identifying 'like terms' is essential. Like terms are terms within an expression that have the exact same variable raised to the same power. For example, in the expression \(2y + y\), both terms are 'like terms' because they include the variable \(y\) raised to the power of 1. Dealing with like terms involves:

• Ensuring the variable part is identical in all terms.
• Checking the exponent on these variables to make sure they match.
• Combining like terms by adding or subtracting their coefficients.

For instance, in step 1 of our solution, the like terms \(2y + y\) were combined by adding their coefficients to form \(3y\). This simplification process is fundamental in algebra, making equations easier to manage and solve.

Exploring Coefficients

Coefficients are the numerical parts that accompany variables in algebraic terms. They are the numbers you multiply by the variable, representing how many times the variable is taken. In the expression \(2y\), the number 2 is the coefficient. It tells us that there are two instances of the variable \(y\). Understanding coefficients involves:

• Recognizing the numbers directly multiplying a variable.
• Adding coefficients when combining like terms.
• Multiplying coefficients during operations involving terms.

For example, in step 2 of our task, \(2y \cdot y = 2y^2\) the coefficient 2 was multiplied by 1 (implied in \(y\)) to maintain the correct quantity of \(y\). Coefficients play a crucial role in accurately simplifying and solving algebraic expressions.

Exponents and Their Role

Exponents indicate how many times a number or variable is multiplied by itself. They are written as small numbers above the variable. This is crucial for correctly handling expressions in algebra. For example, in \(y^2\), the exponent 2 shows that \(y\) is multiplied by itself. When working with exponents:

• Add exponents when you are multiplying like bases, such as \(y \cdot y = y^2\).
• Keep the exponents the same when adding or subtracting like terms.
• Be mindful of rules for negative exponents, although not applicable here.

As seen in our exercise, when \(2y \cdot y\) was processed, we added the exponents (\(1 + 1 = 2\)) to get \(y^2\), indicating two \(y\)s multiplied together. Understanding exponents ensures precision in algebraic simplification.

Working with Negative Numbers

Negative numbers can sometimes be tricky, but they follow specific rules that are not difficult to remember. In algebra, understanding these rules helps avoid common mistakes, especially during operations like multiplication and addition. Key rules are:

• Adding a negative number is the same as subtracting its positive value.
• Subtracting a negative number becomes an addition (e.g., \(5 - (-3) = 8\)).
• Multiplying two negative numbers yields a positive result.

In the expression \(-2y - y\), recognizing both terms as negatives, their coefficients were added (\(-2 - 1 = -3\)). Similarly, \((-2y)(-y)\) involved multiplying two negative coefficients, resulting in a positive product. Understanding how negative numbers interact ensures accuracy in your algebraic computations.