Problem 11 - College Algebra with Modeling and Visualization

Question

Combine like terms whenever possible. $$7 y+9 x^{2} y-5 y+x^{2} y$$

Step-by-Step Solution

Answer

The simplified expression is \(2y + 10x^2y\).

Step 1: Identify Like Terms

In the expression, we identify the terms with similar variables and exponents: \(7y\), \(-5y\), and \(9x^2y + x^2y\). The terms \(7y\) and \(-5y\) are like terms, as are \(9x^2y\) and \(x^2y\).

Step 2: Combine Like Terms - Linear Terms

For the linear terms in \(y\), combine \(7y\) and \(-5y\): \(7y - 5y = 2y\).

Step 3: Combine Like Terms - Quadratic Terms

For the quadratic terms in \(x^2y\), combine \(9x^2y\) and \(x^2y\): \(9x^2y + x^2y = 10x^2y\).

Step 4: Write the Simplified Expression

Combine the results from Step 2 and Step 3 to get the final simplified expression: \(2y + 10x^2y\).

Key Concepts

Combining Like Terms

Combining like terms is a fundamental skill in polynomial simplification. It helps simplify expressions by reducing the number of terms. A good way to recognize like terms is by identifying terms that have the same variables raised to the same powers. Consider the expression from our exercise:

• Terms like \(7y\) and \(-5y\) both involve the variable \(y\). As they have the same exponent of 1 for \(y\), they are considered like terms.
• Similarly, \(9x^2y\) and \(x^2y\) share the same variables, \(x^2\) and \(y\), with their respective powers.

When you spot like terms, combine them by summing their coefficients. For instance, in the exercise, you would compute:

• \(7y - 5y\) to get \(2y\).
• \(9x^2y + x^2y\), knowing that \(x^2y\) is implicitly \(1x^2y\), sums up to \(10x^2y\).

Combining terms in this way ensures the expression remains manageable and is easier to interpret or solve in subsequent steps.

Linear and Quadratic Terms

Understanding the difference between linear and quadratic terms is crucial when tackling algebraic expressions.

• A linear term is a term where the variable is raised to the power of one, such as \(y\), \(2x\), or \(-5y\).
• In contrast, a quadratic term involves the variable squared, such as \(x^2\), \(3x^2\), or \(x^2y\).

In the provided exercise, we deal with both types of terms:
Linear Terms:

• \(7y\) and \(-5y\) are linear because \(y\) is raised to the first power.

Quadratic Terms:

• \(9x^2y\) and \(x^2y\) are quadratic terms. Here \(x\) is squared, adding a degree of complexity to the term.

Recognizing these helps in identifying which like terms can be combined. Linear terms can combine with other linear terms, and quadratic terms combine with other quadratic terms, never with linear ones.

Algebraic Expressions

An algebraic expression combines numbers, variables, and arithmetic operations. It’s like a mathematical sentence needing simplification or evaluation to solve problems.
For example, the expression \(7y + 9x^2y - 5y + x^2y\) involves variables and terms that can be manipulated.

• The goal is to combine terms and simplify the expression.
• Simplification often makes expressions easier to understand and solve.

Important components of algebraic expressions include:

• **Terms**: Individual parts of the expression, divided by addition or subtraction. In our case: \(7y\), \(9x^2y\), \(-5y\), and \(x^2y\).
• **Coefficients**: Numbers multiplying the variables, like 7 in \(7y\) and 9 in \(9x^2y\).
• **Variables**: Symbols representing numbers whose values are not fixed, such as \(x\) and \(y\).

Grasping these basics is essential when starting out with algebra, as they form the foundation for more complex operations and problem-solving techniques.