Problem 66 - Algebra A Combined Function

Question

Evaluate each expression when \(x=-5, y=4,\) and \(t=10\). \(t^{2}-x\)

Step-by-Step Solution

Answer

The value of the expression is 105.

Step 1: Identify the given expression

The expression given is \(t^{2} - x\).

Step 2: Substitute the values

We need to substitute \(t = 10\) and \(x = -5\) into the expression. This will give us \(10^{2} - (-5)\).

Step 3: Calculate the power

Calculate \(10^{2}\), which is \(10 \times 10 = 100\).

Step 4: Apply subtraction

After calculating the power, we have the expression as \(100 - (-5)\). The minus and the negative sign will turn to a plus, so it becomes \(100 + 5\).

Step 5: Final calculation

Perform the final addition \(100 + 5 = 105\).

Key Concepts

Evaluating Expressions

To solve algebraic expressions, evaluating them means finding their value when specific numbers replace the variables. It is a critical skill in algebra that allows us to find specific answers based on given scenarios.
When you evaluate expressions, follow these steps:

• Identify the expression you need to evaluate.
• Substitute the given values for each variable in the expression.
• Perform the necessary mathematical operations, including arithmetic and following the order of operations.

Paying close attention to the signs and operations will help you avoid common mistakes, especially when dealing with negatives and order of operations.

Substitution Method

The substitution method is a straightforward process used in evaluating expressions, which involves replacing variables with specific values. This method is vital because it transforms algebraic expressions into numerical expressions, which are easier to compute.
In our example, the expression is \(t^2 - x\). To solve it using substitution:

• Replace \(t\) with 10 and \(x\) with -5.
• The expression becomes \(10^2 - (-5)\).

This substitution step converts our problem into a simpler arithmetic problem, making it more direct to calculate. Keep in mind that correctly substituting values is key to obtaining the right answer.

Powers and Exponents

When dealing with algebraic expressions, understanding powers and exponents is crucial. Powers are repeated multiplications of the same number. An exponent tells you how many times to multiply the base number by itself.
In the expression \(t^2\), the number 10 is the base, and 2 is the exponent. This means you need to multiply 10 by itself:

• Calculate \(10 \times 10\), which equals 100.

Understanding how to calculate powers is essential for simplifying and solving expressions accurately, as seen in our example.

Integer Arithmetic

Integer arithmetic deals with whole numbers and their operations, such as addition, subtraction, multiplication, and division. It is important because integers include both positive and negative numbers, so understanding how to handle them correctly is key.
In our expression, after calculating \(t^2\) to 100, we had:

• Subtract \(-5\) from 100, expressed as \(100 - (-5)\).
• Subtracting a negative is the same as adding the positive, thus, \(100 + 5 = 105\).

Correctly managing integer operations ensures that calculations reflect accurate results, especially when dealing with negative numbers in expressions.