Problem 3 - Algebra 1

Question

Decide whether the following is an expression, an equation, or an inequality. Explain your decision. $$\left(y^{2}+4\right)-7$$

Step-by-Step Solution

Answer

\((y^{2}+4)-7\) is an expression.

Step 1: Analyze the Given Mathematical Phrase

Look carefully at \((y^{2}+4)-7\). It contains variables y(squared), constants (4,7), and operations (addition and subtraction). However, no 'equal-to' or 'inequality' signs are present.

Step 2: Categorize the Phrase

Considering its features, lack of an equal or inequality sign, this means that \((y^{2}+4)-7\) is a mathematical expression.

Key Concepts

Variables in Expressions

Variables are like the unknowns or placeholders in math. They are often represented by letters such as \(x\), \(y\), or \(z\). In our example, the variable is \(y\). Variables can change value, which makes them very versatile. Imagine variables as empty boxes that can hold any number. In

• \(y^2\): \(y\) is the variable, and squaring it means multiplying \(y\) by itself.

The role of the variable is crucial in expressions because it allows us to create generalizations and solve problems by seeing how different values of the variable affect the outcome. Remember, without a variable, expressions would be just straightforward calculations.

Constants in Expressions

Constants are the numbers that stay the same, no matter what. In our expression, \((y^2 + 4) - 7\), the constants are \(4\) and \(-7\).

• \(4\): Keeps its value in the expression, it does not change.
• \(-7\): Subtracted from the whole expression, it also remains unchanged.

These constants help give shape and structure to the expression, working alongside the variables. They provide a fixed reference, so while the variable part (\(y\)) may change, the constants ensure there is a steady element in the equation.

Operations in Expressions

Operations are the actions that change or combine numbers and variables in expressions. In \((y^2 + 4) - 7\):

• Addition \((+4)\): Adds \(4\) to \(y^2\), creating a sum.
• Subtraction \((-7)\): Subtracts \(7\) from the sum \((y^2 + 4)\).

These operations are what transform the components of expressions and define the relationships between them. Unlike equations or inequalities, expressions only use operations to arrange and modify the components, without making a definitive statement about equality or order. Knowing how operations function helps simplify and evaluate expressions correctly.