Problem 63 - Algebra A Combined Function

Question

Evaluate each expression when \(x=-5, y=4,\) and \(t=10\). \(|x|+2 t-8 y\)

Step-by-Step Solution

Answer

The evaluated expression equals -7.

Step 1: Identify the Expression Components

The expression given to us is \(|x|+2t-8y\). We need to evaluate this expression using the given values of the variables: \(x=-5\), \(y=4\), and \(t=10\).

Step 2: Substitute the Values into the Expression

Replace the variables in the expression with the given values: \(|x|+2t-8y\) becomes \(|-5| + 2(10) - 8(4)\).

Step 3: Evaluate the Absolute Value

Evaluate \(|-5|\). The absolute value of \(-5\) is 5, since absolute value measures the distance from zero. So, we have 5.

Step 4: Multiply and Simplify

Next, calculate \(2(10)\) which gives 20, and \(8(4)\) which gives 32. Update the expression to be 5 + 20 - 32.

Step 5: Perform the Addition and Subtraction

Add and subtract the numbers from left to right: \(5 + 20 = 25\) and then \(25 - 32 = -7\). The result is \(-7\).

Step 6: State the Final Solution

The evaluated expression \(|-5| + 2(10) - 8(4)\) equals \(-7\).

Key Concepts

Absolute Value

The absolute value of a number is a concept that helps in understanding the non-negative magnitude of that number. It tells us how far a number is from zero on the number line, regardless of direction. For any real number, the absolute value function is denoted by vertical bars. For instance, the absolute value of \( -5 \) is denoted as \( |-5| \).
- Absolute value is always zero or positive.- It is expressed as the number itself, if it's positive, or its opposite if it's negative.For example, in our exercise, the calculation of \( |-5| \) results in 5, since we're concerned only with the number's distance from zero, not its direction. This makes it particularly useful in various mathematical operations, as it standardizes values by disregarding negative signs.

Substitution in Algebra

Substitution in algebra is the process of replacing variables in expressions with their given values, allowing us to evaluate the expressions numerically.
- It simplifies an expression into a numerical problem.- Using known values helps solve algebraic expressions effectively.In our exercise, the expression given was \(|x| + 2t - 8y\). We substituted the given values: \(x = -5\), \(y = 4\), and \(t = 10\) into the expression. This turns the abstract expression into concrete numbers, like so: \(|-5| + 2(10) - 8(4)\). Mastering substitution is pivotal in many algebraic operations and logical problem-solving.

Order of Operations

The order of operations is a foundational principle in mathematics, ensuring consistency and clarity in solving expressions. Without a clear order, mathematical expressions could yield different results, making them unreliable. To avoid confusion, it's crucial to adhere to the sequence of operations often remembered by the acronym PEMDAS:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our example, after substituting the values, the expression was \(|-5| + 2(10) - 8(4)\). Here, we follow the order: evaluate \(|-5|\), perform multiplications \(2(10) = 20\) and \(8(4) = 32\), then proceed with addition and subtraction. Correctly following the order, we compute \(5 + 20 = 25\) and subtract \(32\) to arrive at \(-7\). Respecting the order of operations is key to accurate calculations.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and computational symbols like addition and subtraction. They form the basis of algebra and help in formulating real-world problems into a solvable format.
- Expressions can range from simple to complex.- They often represent unknown quantities.The expression in the exercise \(|x| + 2t - 8y\) involves both numbers and operations acting on variables. It highlights how we can create expressions that describe different relations and conditions using algebra.By substituting given values into the variables, such expressions transform into a numeric calculation,allowing us to solve or evaluate as per the given conditions. Mastering algebraic expressions, and their formulation, is essential for advanced problem-solving.