Problem 55 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The expression evaluates to \(-3\).

Step 1: Substitute the Given Values

Start by substituting the values for variables into the expression. Given that \( x = -2 \), \( y = 6 \), and \( z = 5 \), substitute these into the expression \( 5x + 2y - z \). This becomes \( 5(-2) + 2(6) - 5 \).

Step 2: Calculate Each Term Separately

Calculate each term individually before summing them. First, evaluate \( 5(-2) \), which equals \(-10\). Then evaluate \( 2(6) \), which equals \(12\). Finally, the term \(-z\) is \(-5\).

Step 3: Combine the Results

Add the three results from Step 2 together. Combine \(-10\), \(+12\), and \(-5\): \(-10 + 12 - 5\).

Step 4: Simplify the Expression

Simplify the results from Step 3. First, add \(-10\) and \(12\) to get \(2\). Then subtract \(5\) from \(2\) to get \(-3\).

Key Concepts

Expression Evaluation

Expression evaluation involves determining the value of an algebraic expression, given specific values for its variables. The first step in expression evaluation is replacing each variable with its given value. This process is known as variable substitution.

• Identify the values of variables, like in our example where \( x = -2 \), \( y = 6 \), and \( z = 5 \).
• Substitute these values directly into the expression, which in our case is \( 5x + 2y - z \).

After substitution, calculate the results for each part of the expression. It’s crucial to pay attention to the mathematical operations involved, such as multiplication, addition, and subtraction, and follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).

Algebraic Expressions

Algebraic expressions are mathematical phrases combining numbers, variables, and operation symbols. They can be as simple as a single number or variable, or they can involve multiple terms and operations.

• A term is any combination of constants and/or variables multiplied together. For example, in the expression \( 5x + 2y - z \), "\( 5x \)", "\( 2y \)", and "\( -z \)" are its terms.
• It's important to understand that variables in these expressions stand for unknown values that can be changed or substituted.

By manipulating these algebraic expressions, one can solve equations, construct formulas, and model real-world scenarios. Understanding how to work with these expressions is foundational to algebra and higher mathematics.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their most basic form. This often involves combining like terms and performing operations. In our example, after evaluating and substituting the given values into the expression, it's important to simplify the result for clarity.

• Begin by calculating individual terms, such as \( 5 \times (-2) = -10 \), \( 2 \times 6 = 12 \), and \(-z = -5\).
• Combine these calculated terms to simplify further: first, add \(-10\) and \(12\) to get \(2\), and then subtract \(5\), resulting in \(-3\).

Simplifying expressions is essential for understanding and communicating mathematical ideas clearly. It ensures you can see the most streamlined version of an expression, making further calculations or interpretations easier.