Problem 52 - Algebra 2

Question

Evaluate each expression if \(x=-2, y=6,\) and \(z=5\) $$ x+y+z $$

Step-by-Step Solution

Answer

9

Step 1: Substitute the given values into the expression

Substitute the values of the variables into the expression. We have \( x = -2 \), \( y = 6 \), and \( z = 5 \). The expression \( x + y + z \) becomes \( -2 + 6 + 5 \).

Step 2: Perform the addition

Start by adding the first two numbers: \( -2 + 6 = 4 \). Then add the third number: \( 4 + 5 = 9 \).

Key Concepts

Substitution Method

In algebra, the substitution method is a powerful tool that replaces variables in an algebraic expression with given values to simplify and evaluate it. Let's see how this works using the given exercise. We are provided with an expression:

• The expression is: \( x + y + z \)
• The given values are: \( x = -2 \), \( y = 6 \), and \( z = 5 \)

So, to evaluate the expression, we substitute these values into our original expression. By placing each numeric value in place of its corresponding variable, the expression transforms into actual numbers we can calculate: \[ -2 + 6 + 5 \]Now, the algebraic expression is ready for the next steps of arithmetic operations. This method helps in simplifying and solving various algebraic problems seamlessly by making substitution an early step in problem-solving.

Addition of Integers

Once substitution is completed, the expression turns into a straightforward integer addition problem. Understanding how to add integers is essential:

Let's start by adding the first two integers, \( -2 \) and \( 6 \). Since \( -2 \) and \( 6 \) are different in sign, we will subtract the absolute value of \(-2\) from \(6\):

• \( -2 + 6 = 4 \)

Next, we take the sum \( 4 \) and add the last integer \( 5 \):

• \( 4 + 5 = 9 \)

The result of \( 9 \) is obtained by continuously applying addition rules to the substituted values. Remember, the key is aligning numbers correctly, considering their signs, and performing operations step-by-step, which ensures accuracy in results.

Algebraic Expressions

An algebraic expression is a combination of numbers, variables, and arithmetic operations. These expressions represent real-world situations or abstract mathematical ideas. In the given exercise, we have a simple expression combining the variables \( x, y,\) and \( z \) with addition.

Here's a quick breakdown of an algebraic expression:

• Variables: Symbols that stand in place of numbers (e.g., \( x, y, z \))
• Constants: Numbers without variables, fixed values in the expression
• Operators: Arithmetic symbols like \( +, -, \, \times, \, \div \)

Algebraic expressions can be evaluated, simplified, or used to form equations and evaluate solutions. You create a bridge from variable-based scenarios to numeric results using the substitution method, as showcased. Understanding how to manipulate and calculate these expressions is key to succeeding in algebra and further math courses.