Problem 63 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The expression evaluates to 132.

Step 1: Identify the Expression

The expression we need to evaluate is: \[ x^{2} - 3y + x \] Our task is to replace the variables \( x \), \( y \), and \( z \) with the given values, where \( x = 12 \), \( y = 8 \), and \( z = 4 \). Although \( z \) is provided, it is not needed in this expression.

Step 2: Substitute Values

Substitute the given values into the expression: \[ 12^{2} - 3(8) + 12 \] Replace each variable with its corresponding value: \( x = 12 \), \( y = 8 \).

Step 3: Calculate Each Term

First, calculate \( 12^2 \):\[ 12^2 = 144 \]Next, calculate \( 3 \times 8 \):\[ 3 imes 8 = 24 \]Now plug these values back into the expression: \[ 144 - 24 + 12 \]

Step 4: Perform Addition and Subtraction

With all terms calculated, perform addition and subtraction:First, subtract 24 from 144:\[ 144 - 24 = 120 \]Then, add 12 to the result:\[ 120 + 12 = 132 \]

Key Concepts

Substitution in Algebra

Substitution in algebra is a fundamental concept that involves replacing variables in an expression with specific values. This technique helps us simplify and solve expressions or equations.

When performing substitution, we need to carefully identify the variables and replace them with their corresponding numerical values. In our original exercise, we have the expression \( x^{2} - 3y + x \). For substitution:

• Replace \( x \) with 12
• Replace \( y \) with 8

This means, whenever you see an \( x \), you substitute it with 12, and wherever you see a \( y \), you replace it with 8. This effectively transforms the algebraic expression into a numeric computation that can be easily solved.

It's also important to note that the variable \( z \) is given but not used in this particular expression. Always focus on the variables that appear in your specific equation or expression.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations such as addition, subtraction, multiplication, and division.

An algebraic expression can be seen as a recipe consisting of several components:

• Variables like \( x \), \( y \), and \( z \) represent unknown values. They are placeholders that allow the expression to be reused with different numbers.
• Coefficients are numbers used to multiply a variable. In the expression \( -3y \), the number -3 is the coefficient of \( y \).
• Constants are numbers on their own, such as \( x^{2} \) and \( x \) in our example.

When evaluating an expression, understanding each component helps in simplifying and solving it accurately. By substituting known values for variables, the expression becomes a straightforward arithmetic problem.

Step-by-Step Solution

Solving algebraic expressions step-by-step is a systematic approach to ensure accuracy and understanding. Let's break down the example of \( x^{2} - 3y + x \):

**Step 1: Identify the Expression**
First, clearly identify what you need to solve. This means recognizing all parts of the expression and what's being asked.

**Step 2: Substitution**
Substitute the values given in the problem into the expression. Replace \( x \) with 12 and \( y \) with 8. This gives us the numerical expression \( 12^2 - 3 \times 8 + 12 \).

**Step 3: Calculate Each Term**
Break down the expression into parts and calculate individually:

• Compute \( 12^2 \) to get 144.
• Compute \( 3 \times 8 \) to get 24.

Plug these back into the expression: \( 144 - 24 + 12 \).

**Step 4: Perform the Operations**
Carry out the operations step by step:

• First, subtract 24 from 144, resulting in 120.
• Then, add 12 to 120 to arrive at the final answer of 132.

Taking it one step at a time ensures you handle each part correctly, minimizing mistakes and enhancing understanding.