Problem 2 - Algebra 2

Question

Evaluate each expression if \(x=4, y=-2,\) and \(z=3.5\) \(x+(y-1)^{3}\)

Step-by-Step Solution

Answer

The value of the expression is -23.

Step 1: Identify Variables

Identify the given values for the variables in the expression. Here, we have:- \( x = 4 \)- \( y = -2 \)- \( z = 3.5 \)

Step 2: Substitute Values into Expression

The expression is \( x + (y - 1)^{3} \). Substitute the given values for \(x\) and \(y\):\[ 4 + ((-2) - 1)^{3} \]

Step 3: Simplify Inside the Parentheses

Calculate \( y-1 \) using \( y = -2 \):\[ (-2) - 1 = -3 \]

Step 4: Cube the Result

Now, cube the result from the previous step:\[ (-3)^{3} = (-3) \times (-3) \times (-3) = -27 \]

Step 5: Evaluate the Entire Expression

Substitute the cubed result back into the expression and evaluate:\[ 4 + (-27) = 4 - 27 = -23 \]

Key Concepts

Substitution in Algebra

Substitution in algebra is like plugging values into a formula to solve a problem. It's an essential method used to find a specific answer when dealing with variables, which are symbols representing numbers. Variables allow us to create general rules and formulas, but to get a specific result, we need to substitute the values given.

In the context of the exercise, we have specific numbers assigned:

• For \( x, \) the value is \( 4 \)
• For \( y, \) the value is \( -2 \)
• For \( z, \) the value is \( 3.5 \)

We then substitute these values into the algebraic expression provided. This way, the unknowns disappear, replaced by actual numbers, making it possible to solve the problem.

Think of substitution as replacing the placeholders in a recipe with real ingredients. Without replacing the placeholders, you wouldn’t know the actual ingredients needed to make the dish!

Polynomial Expressions

Polynomial expressions are algebraic expressions that consist of variables and constants combined using addition, subtraction, and multiplication. These expressions can have one or more terms, and the example given is simple yet powerful because it involves a term raised to a power.

In our exercise, the expression \(x + (y - 1)^{3}\) is a polynomial expression. It is made up of an addition operation where one portion is a cubic term \((y - 1)^{3}\).

• This expression shows us that a polynomial can have terms of varying degrees, which are essentially the powers to which the variables are raised.
• The degree of the polynomial is determined by the term with the highest power, which in this case is 3.

Understanding polynomial expressions is crucial because they appear in many algebraic applications, from simple arithmetic equations to complex calculus problems. Once we substitute the values, a polynomial becomes a simple arithmetic problem ready to be solved.

Order of Operations

Order of operations is a fundamental concept in mathematics used to determine the correct sequence of steps when simplifying mathematical expressions. Without a standard order, different people might solve the same expression differently. The commonly used rule is PEMDAS:

• P: Parentheses first
• E: Exponents (i.e., powers and roots, etc.)
• M/D: Multiplication and Division (left to right)
• A/S: Addition and Subtraction (left to right)

In our expression \( x + (y-1)^{3} \), the order of operations guides us in simplifying the expression:

• Firstly, we deal with the parentheses. This means we calculate \( y-1 \), so \((-2) - 1 = -3 \).
• Next, handle the exponent by cubing \(-3\). Hence, \((-3)^{3} = -27 \).
• Finally, perform the addition: \( 4 + (-27) = -23 \).

Understanding and applying the correct order of operations ensure that we solve algebraic expressions accurately, resulting in the correct answer. Remember, following the order of operations is key to consistently arriving at the right solution.