Problem 54 - Algebra 2

Question

Evaluate each expression if \(x=-2, y=6,\) and \(z=5\) $$ -x+4 y-2 z $$

Step-by-Step Solution

Answer

The value of the expression is 16.

Step 1: Identify and Substitute Values

Begin by identifying the variables in the expression: \(x\), \(y\), and \(z\). Substitute the given values: \(x = -2\), \(y = 6\), and \(z = 5\) into the expression \(-x + 4y - 2z\).

Step 2: Calculate -x

Substitute \(x = -2\) into \(-x\), which gives \(-(-2) = 2\).

Step 3: Calculate 4y

Substitute \(y = 6\) into \(4y\), which gives \(4 \times 6 = 24\).

Step 4: Calculate -2z

Substitute \(z = 5\) into \(-2z\), which gives \(-2 \times 5 = -10\).

Step 5: Combine the Results

Add the results from the previous steps: \(2 + 24 - 10\).

Step 6: Simplify the Expression

Calculate the result: \(2 + 24 - 10 = 16\).

Key Concepts

Substitution

Substitution is one of the key skills in algebra required for solving expressions. It involves replacing variables within an algebraic expression with known numerical values.
For the expression \(-x + 4y - 2z\), begin by recognizing each variable—\(x\), \({y}\), and \({z}\). Knowing the values \(x = -2\), \(y = 6\), and \(z = 5\), you substitute these into the expression.

• Replace \(x\) with \(-2\)
• Replace \(y\) with \(6\)
• Replace \(z\) with \(5\)

This step leads to the transformation of the expression into a numerical computation, ready for further simplification through arithmetic operations.

Evaluation of Expressions

Evaluation of expressions means replacing the variables with their specific numerical values, followed by calculations to get a single number. Once substitution is complete, you continue by performing arithmetic operations on the resulting numeric expression.
Let's examine how each term in \(-x + 4y - 2z\) is evaluated:

• Calculate \(-x\): substitute and compute \(-(-2) = 2\)
• Calculate \(4y\): substitute and compute \(4 \times 6 = 24\)
• Calculate \(-2z\): substitute and compute \(-2 \times 5 = -10\)

These calculations give numbers that replace each algebraic term, allowing the expression to be fully evaluated.

Arithmetic Operations

Arithmetic operations are the foundational operations that include addition, subtraction, multiplication, and division. In evaluating expressions, after substitution and calculation, the results are combined using these operations to achieve a final simplified number.
For the expression \(-x + 4y - 2z\) after substitution, the arithmetic operation steps are clearly needed:

• Add \(2 + 24\) from the solved terms \(-x\) and \(4y\)
• Subtract \(10\) which is the result of \(-2z\) from \(26\)

The combination of these results gives you the expression's final value, simplifying it to \(-x + 4y - 2z = 16\). Arithmetic operations complete the process of finding the numeric value of algebraic expressions.