Problem 56 - Algebra A Combined Function

Question

Evaluate each expression when \(x=1, y=3,\) and \(z=5 .\) $$ |5 z-2 y| \quad $$

Step-by-Step Solution

Answer

Evaluating the expression gives 19.

Step 1: Substitute the given values

Start by substituting the given values for the variables into the expression. We know:\[x=1, y=3, z=5\]The expression is:\[|5z-2y|\]Substitute these values into it:\[|5(5)-2(3)|\]

Step 2: Simplify the expression

Now, perform the multiplication inside the absolute value:\[|25 - 6|\]This involves multiplying 5 by 5 to get 25, and 2 by 3 to get 6.

Step 3: Calculate inside the absolute value

Next, subtract 6 from 25:\[|19|\]The result inside the absolute value is 19.

Step 4: Apply the absolute value

Since 19 is a positive number, the absolute value does not change it. So:\[|19| = 19\]

Key Concepts

Absolute Value

Absolute value is a fundamental concept in mathematics, representing the distance of a number from zero on a number line. Simply put, it tells us how "far" a number is from zero, regardless of its direction. For example, both 3 and -3 have an absolute value of 3, because they are 3 units away from zero.

When examining the absolute value, we use the notation with vertical bars, like \(|x|\). If the number inside the bars is positive or zero, the absolute value remains the same. If it's negative, we simply drop the negative sign to obtain the absolute value.

In the original exercise, after calculations within the absolute value symbol \(|5z-2y|\), we ended up with \(|19|\). Since 19 is already positive, the absolute value doesn't change it. This makes the final answer straightforward: just 19!

Substitution Method

The substitution method helps simplify expressions by replacing variables with their given values. It allows us to transform an algebraic expression with unknowns into a numerical expression that we can compute easily. This technique is particularly useful when dealing with specific scenarios or conditions provided in a problem.

In the given exercise, we needed to evaluate \(|5z - 2y|\) when \(|x=1, y=3, z=5|\). Using the substitution method, we replaced \(z\) with 5 and \(y\) with 3, converting the expression to \(|5(5) - 2(3)|\). By systematically substituting each variable, we reduced the complexity of the initial problem, paving the way for direct calculations.

Arithmetic Operations

Arithmetic operations are the basic mathematical procedures used to solve expressions, including addition, subtraction, multiplication, and division. These operations follow the order of operations (PEMDAS/BODMAS), helping us decide which calculations to do first.

In this exercise, after substitution, we performed a series of arithmetic operations within the absolute value expression. Specifically, we:

• Multiplied: Calculated \(5 \times 5\) to get 25, and \(2 \times 3\) to get 6.
• Subtracted: 25 minus 6 resulted in 19.

Mastering these operations is crucial, as they form the foundation of more complex mathematical solving techniques.

Simplification in Algebra

Simplification in algebra is about reducing an expression to its simplest form. This often involves performing arithmetic operations, combining like terms, or applying properties like the distributive property. The ultimate goal is to make the expression as straightforward as possible.

In the exercise, once we substituted and performed arithmetic operations, we simplified \(|5(5) - 2(3)|\) to \(|19|\). Simplification made it easier to apply the absolute value rule, ultimately finding the answer with minimal confusion.

By breaking down each expression step-by-step, learning simplification techniques ensures a smooth, logical flow of problem-solving, vital for tackling more challenging algebraic problems in the future.