Problem 64 - Algebra A Combined Function

Question

Evaluate each expression when \(x=-5, y=4,\) and \(t=10\). \(|y|+3 x-2 t\)

Step-by-Step Solution

Answer

The expression evaluates to \(-31\).

Step 1: Substitute Values

Start by replacing each variable in the expression with its given value. The expression is \(|y|+3x-2t\). Substitute \(y=4\), \(x=-5\), and \(t=10\) into it. So it becomes \(|4| + 3(-5) - 2(10)\).

Step 2: Simplify Absolute Value

The first term is the absolute value of \(y\). Since \(y=4\), the absolute value \(|4|\) is simply \(4\).

Step 3: Compute Multiplications

Calculate each multiplication in the expression. For \(3(-5)\), we have \(3 imes -5 = -15\). Similarly, for \(2(10)\), we have \(2 imes 10 = 20\).

Step 4: Combine Terms

Now, substitute the computed values back into the expression. It becomes: \(4 + (-15) - 20\).This is equivalent to \(4 - 15 - 20\).

Step 5: Execute Arithmetic

Perform the arithmetic operations by following the order from left to right. Start with \(4 - 15 = -11\), and then compute \(-11 - 20 = -31\).

Key Concepts

Understanding Absolute Value

Absolute value is a fundamental concept in mathematics. It represents the distance of a number from zero on a number line, regardless of direction. When dealing with absolute value, it is important to remember:

• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is the number without its sign.

For example, the absolute value of -5 is 5, and the absolute value of 4 is 4. In the original exercise, we encountered \(|y|\), which means we need to find the absolute value of \(y\). Given \(y = 4\), the absolute value \(|4|\) is simply 4. This step ensures that the value taken into the expression remains non-negative.

Algebraic Substitution Simplified

Algebraic substitution is replacing variables with their given values to simplify an expression. It's like swapping puzzle pieces with their exact fits. In the expression \( |y| + 3x - 2t \), we're told what each piece represents:

• \(x = -5\)
• \(y = 4\)
• \(t = 10\)

We simply substitute each of these variables with their values. This turns the expression into \( |4| + 3(-5) - 2(10) \). Substitution helps us transition from a theoretical expression to a practical calculation.

Navigating Arithmetic Operations

Arithmetic operations include addition, subtraction, multiplication, and division. In evaluating the expression, you're mainly dealing with these operations. It is significant to handle each operation carefully to maintain accuracy. Here's what to note:

• Multiplication: Compute \(3 imes -5 = -15\) and \(2 \times 10 = 20\).
• Subtraction and Addition: Combine the results carefully: \(4 + (-15) - 20\) simplifies to \(4 - 15 - 20\).

Always take care with signs when multiplying or adding numbers. Negative and positive numbers change equations considerably!

The Role of Order of Operations

Order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial to solving mathematical expressions correctly. It tells us the sequence in which operations must be performed to reach the right result.For our expression \(4 + (-15) - 20\), we follow this sequence:

• Solve any operations inside parentheses. Here, \(4\) stands alone.
• Perform all multiplications and divisions, from left to right, as shown previously.
• Lastly, perform addition and subtraction from left to right: start with \(4 - 15\) which equals \(-11\), followed by \(-11 - 20\) which gives \(-31\).

Strict adherence to the order of operations ensures consistency and correctness in solving expressions.