Problem 85 - Elementary and Intermediate Algebra

Question

Evaluate each expression, for \(x=3, y=-2,\) and \(z=-4\) See Example 10. $$ (x+y)^{2}-|z+y| $$

Step-by-Step Solution

Answer

The evaluated expression is -5.

Step 1: Substitute the Values

First, replace the variables in the expression \((x+y)^{2} - |z+y|\) with the given values: \(x=3\), \(y=-2\), and \(z=-4\). This gives us: \((3-2)^{2} - |-4 - 2|\).

Step 2: Simplify the Expression Inside the Parentheses

Calculate the simplified form of the expression inside the parentheses: \(3-2=1\). Thus, the expression becomes \(1^2 - |-4-2|\).

Step 3: Calculate the Power

Evaluate the square of the result from step 2: \(1^2 = 1\). Now the expression is \(1 - |-4-2|\).

Step 4: Simplify the Absolute Value

First, simplify the expression inside the absolute value: \(-4 - 2 = -6\). Then, find the absolute value, which is \|-6\| = 6. Thus, the expression is now \(1 - 6\).

Step 5: Perform the Final Calculation

Subtract the result from the absolute value from the squared result: \(1 - 6 = -5\).

Key Concepts

Substitution Method

The substitution method is a fundamental concept in algebra used to replace variables with specific values in an expression or equation. It allows you to "test out" an expression using real numbers.
Let's see how this method works. Here, given the expression \((x+y)^{2} - |z+y|\), we substitute the provided values: \(x=3\), \(y=-2\), and \(z=-4\).

• Substitute \(x=3\) and \(y=-2\) into the first part: \((3+(-2))^{2}\).
• Similarly, substitute \(z=-4\) and \(y=-2\) into the second part: \(-4 + (-2)\).

After substitution, the expression becomes \((3-2)^{2} - |-4-2|\).
This approach helps in simplifying the expression, making complicated algebraic expressions easy to evaluate.

Order of Operations

The Order of Operations is a set of rules used to ensure that expressions are evaluated in a consistent manner. When evaluating any mathematical expression, follow these steps:

• Parentheses - Perform operations inside parentheses first.
• Exponents - Solve any exponents (powers or roots) next.
• Multiplication and Division - From left to right.
• Addition and Subtraction - From left to right.

For the expression \((3-2)^{2} - |-4-2|\), follow these steps:
1. Inside the parentheses: Calculate \(3 - 2\) resulting in \(1\).
2. Solve the exponent: Calculate \(1^2 = 1\).
3. Simplify the absolute value (more on this in the next section).
Following these rules will always lead to the correct solution. They help manage the complexity in expressions by providing a clear order.

Absolute Value

The absolute value of a number refers to its distance from zero on the number line, without considering direction. It's always a non-negative number.
If you have \-\([expression]\)\, you need to ignore the sign and consider only the magnitude.
In our example, inside the absolute value \(|z + y|\), substitute and simplify to \(|-4 - 2|\); which results in \(|-6|\).

• The absolute value of \(-6\) is \(6\).

This step transforms the expression to \(1 - 6\).
Understanding absolute value is vital as it can change the outcome in many expressions and equations by simplifying them to a positive number regardless of the initial negative input.

Simplifying Expressions

Simplifying an expression means rewriting it in the simplest or most efficient form by performing all possible operations.
From our expression, we've substituted values and performed operations: \((3-2)^{2} - |-4-2|\) becomes \(1^2 - 6\).

• First, we calculated within the parentheses: \(3-2 = 1\).
• Then, we dealt with the power: \(1^2 = 1\).
• Finally, subtract the result of the absolute value: \(1 - 6\).

This consistent approach results in \(-5\), showing all operations applied sequentially and correctly.
Simplifying expressions helps in making them easier to understand and solve, often needed for quick and accurate calculations in algebra.