Problem 57 - Algebra 2

Question

Evaluate each expression if \(x=-2, y=6,\) and \(z=5\) $$ -2 x-3 y+2 z $$

Step-by-Step Solution

Answer

The evaluated expression is -4.

Step 1: Substitute the Values

Replace the variables in the expression with the given values. The expression is \(-2x - 3y + 2z\). Substitute \(x = -2\), \(y = 6\), and \(z = 5\). The expression becomes \(-2(-2) - 3(6) + 2(5)\).

Step 2: Multiply Coefficients by Numbers

Calculate the products of the numbers and their coefficients. Begin with \(-2(-2)\), which becomes \(4\). Then, \(-3(6)\) becomes \(-18\). Lastly, \(2(5)\) equals \(10\).

Step 3: Combine the Values

Add the results from the previous calculations together: \(4 - 18 + 10\).

Step 4: Simplify the Expression

Perform the addition and subtraction: \(4 + 10 = 14\), then \(14 - 18 = -4\).

Key Concepts

Variable Substitution

In algebra, expressions often include one or more variables, which are symbols representing unknown values. To evaluate an algebraic expression, you must first substitute these variables with specific known values. This process is called **variable substitution**. For example, if you have an expression like \[ -2x - 3y + 2z \]and you're given that \( x = -2, y = 6, \) and \( z = 5 \), you would replace each variable in the expression with its corresponding number:

• Replace \( x \) with \(-2\),
• Substitute \( y \) with \(6\),
• Swap \( z \) with \(5\).

This changes the original expression into another expression with numbers only, \[ -2(-2) - 3(6) + 2(5) \]allowing for further evaluation. It’s crucial to ensure each substitution respects the expression’s original structure to maintain accuracy.

Arithmetic Operations

Arithmetic operations are the foundation of simplifying and solving mathematical expressions. They include addition, subtraction, multiplication, and division. In the given expression, \[ -2(-2) - 3(6) + 2(5) \], we focus primarily on multiplication, addition, and subtraction.

• First, perform the multiplication operations: calculate \(-2 \times -2\), \(-3 \times 6\), and \(2 \times 5\).
• This sequence helps break down the larger problem into manageable steps.

Multiplication takes precedence over addition and subtraction, meaning you tackle these parts first before moving on to the other operations.

Coefficients

In algebraic expressions, coefficients are numbers placed directly in front of variables. They indicate how many times the variable is counted in the expression. For instance, in the expression \[ -2x - 3y + 2z \],the coefficients are:

• \(-2\) for \(x\),
• \(-3\) for \(y\), and
• \(2\) for \(z\).

During evaluation, these coefficients are multiplied by the variable's actual value after substitution. So, to evaluate the expression, conceptually, you are performing these calculations:

• \(-2 \times -2\),
• \(-3 \times 6\), and
• \(2 \times 5\).

Each product reflects how the coefficient scales the corresponding variable.

Simplification of Expressions

Once you have substituted the variables and performed the arithmetic operations, the next step is to simplify the expression. **Simplification** involves combining like terms or further processing of the numeric operations until you reach the simplest form of the expression. Here's how it works with our example \[ -2(-2) - 3(6) + 2(5) \]:

• Compute each multiplication: \(-2 \times -2 = 4\), \(-3 \times 6 = -18\), \(2 \times 5 = 10\).
• Add and subtract these results: perform \(4 + 10\) to get \(14\).
• Finally, subtract \(18\) from \(14\), resulting in \(-4\).

This sequence ensures you've accurately interpreted the expression's value by systematically combining the results of your arithmetic operations.