Problem 64 - Algebra A Combined Function

Question

Evaluate each expression when \(x=12, y=8,\) and \(z=4\). $$ y^{2}-3 x+y $$

Step-by-Step Solution

Answer

The value of the expression is 36.

Step 1: Substitute Given Values

Start by substituting the values given for \(x\), \(y\), and \(z\) into the expression \(y^2 - 3x + y\). Here, \(x = 12\), \(y = 8\), and \(z = 4\). The expression becomes: \(8^2 - 3(12) + 8\).

Step 2: Calculate the Square

Proceed to calculate the square of \(y\), which is \(8^2\). So, we have: \(64 - 3(12) + 8\).

Step 3: Multiply and Simplify

Now, multiply \(3\) by \(x = 12\), which gives \(36\). The expression now is: \(64 - 36 + 8\).

Step 4: Perform Addition and Subtraction

Finally, perform the addition and subtraction operations sequentially: first subtract \(36\) from \(64\) to get \(28\), and then add \(8\), giving a total of \(36\).

Key Concepts

Substitution in Expressions

Substitution in mathematical expressions is a process where specific values are replaced in place of the corresponding variables. When we have an algebraic expression like \(y^2 - 3x + y\), each letter represents a different variable or unknown quantity.
To evaluate the expression given specific values, follow this simple substitution method:

• Identify the variables present in the expression. In this case, the variables are \(x\), \(y\), and \(z\).


• Replace each variable with its given value. For example: replacing \(x\) with \(12\), \(y\) with \(8\), and \(z\) with \(4\).


• Re-evaluate the expression with these substitutions. This results in an updated expression like \(8^2 - 3(12) + 8\).

Substituting values into expressions paves the way to calculate results accurately and is an essential skill in solving algebraic problems.

Order of Operations

The order of operations is crucial when evaluating expressions involving multiple arithmetic operations. Without following a standardized order, you might end up with incorrect results.
The standard sequence often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) ensures uniformity and accuracy.
Here's how to apply it:

• First, evaluate any expressions inside parentheses or brackets.


• Next, calculate exponents or powers, such as \(8^2\).


• Then, perform all multiplication and division as they appear from left to right in the expression.


• Finally, handle all addition and subtraction, also from left to right.

Applying the rule to our expression \(8^2 - 3(12) + 8\), you'll see:

• First, \(8^2 = 64\).


• Then, \(3 \times 12 = 36\).


• Lastly, subtract and add in order: \(64 - 36 + 8 = 36\).

The order of operations removes any ambiguity, ensuring everyone arrives at the same result.

Arithmetic Operations

Arithmetic operations, the basic building blocks in mathematics, include addition, subtraction, multiplication, and division. These are the fundamental tools we use to evaluate expressions once variables have been substituted and the correct order of operations is applied.
Let's clarify each operation using our original expression:

• Addition: Combines numbers to give a sum, seen when adding \(8\) to \(-28\) yielding \(36\).


• Subtraction: Finds the difference between numbers, such as subtracting \(36\) from \(64\) to result in \(28\).


• Multiplication: Involves scaling one number by another, exemplified by \(3 \times 12 = 36\).


• Division: Although not needed in the current expression, it typically entails distributing one value into equal parts.

Mastering these operations is crucial for efficiently solving any real or theoretical number problems in mathematics. Familiarity and practice with these operations enhance your problem-solving capabilities.