Problem 14 - Algebra 2

Question

Evaluate each expression if \(w=6, x=0.4, y=\frac{1}{2},\) and \(z=-3\). \(5 x+2 z\)

Step-by-Step Solution

Answer

The value of the expression is -4.0.

Step 1: Substitute the Given Values

Start by substituting the values given for each variable in the expression. Here we substitute \(x = 0.4\) and \(z = -3\) into the expression \(5x + 2z\). Thus, the expression becomes \(5(0.4) + 2(-3)\).

Step 2: Calculate the Multiplication Terms

Next, calculate each multiplication term in the expression. For the first term: \(5 \times 0.4 = 2.0\). For the second term: \(2 \times (-3) = -6\). Now, the expression is \(2.0 - 6\).

Step 3: Perform the Addition/Subtraction

Now, perform the subtraction in the expression. Calculate \(2.0 - 6\) which results in \(-4.0\).

Key Concepts

Substitution

In algebra, substitution is the initial and vital step in evaluating expressions. It involves replacing the variables in an expression with the values provided.

For the expression given in the exercise, we need to substitute the values of the variables, specifically for variable \(x\) and \(z\):

• \(x = 0.4\)
• \(z = -3\)

This transforms the original expression \(5x + 2z\) into a numerical expression: \(5(0.4) + 2(-3)\).

Substitution is important because it converts abstract variables into concrete numbers, making the problem easier to solve.

Multiplication

After substitution, the next step is to handle multiplication. This involves multiplying the coefficients by their respective substituted numerical values.

For our expression:

• The term \(5x\) becomes \(5 \times 0.4\), which results in \(2.0\).
• The term \(2z\) converts into \(2 \times (-3)\), giving \(-6\).

Multiplication simplifies the expression by dealing with individual terms one at a time. Remember, multiplication is performed before addition or subtraction, according to the rules of operations (PEMDAS/BODMAS).

Addition and Subtraction

Once you have the multiplication results, apply addition or subtraction as indicated in the new expression. The current expression is \(2.0 - 6\).

Here’s how to proceed:

• Start with the first number \(2.0\).
• Subtract \(6\) from \(2.0\), resulting in \(-4.0\).

Adding and subtracting are operations that integrate all the multiplied numbers into a final answer. They are the final touch in simplifying expressions.

Variables

Variables are symbols often used to represent unknown quantities in mathematics. In the given expression, \(x\) and \(z\) are variables.

Each variable can adopt different values depending on the context or specific conditions given in a problem.

Understanding variables is crucial because:

• They allow flexibility and generalization in creating formulas and functions.
• They are placeholders that can be substituted with specific numbers to solve problems.

When evaluating expressions, correct handling of variables through substitution will directly affect the outcome of the solution.