Problem 13 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The value of the expression is 5.3.

Step 1: Substitute the Given Values

Start by substituting the given values into the expression: \(w = 6\), \(x = 0.4\), \(y = \frac{1}{2}\). The expression \(w - 3x + y\) becomes: \[6 - 3(0.4) + \frac{1}{2} \].

Step 2: Multiply

Multiply \(3\) by \(0.4\) in the expression:\[3 \times 0.4 = 1.2\].So the expression now becomes \(6 - 1.2 + \frac{1}{2}\).

Step 3: Perform Subtraction

Next, subtract \(1.2\) from \(6\):\[6 - 1.2 = 4.8\].The expression simplifies to \(4.8 + \frac{1}{2}\).

Step 4: Add the Remaining Terms

Finally, add \(\frac{1}{2}\) to \(4.8\):Convert \(\frac{1}{2}\) to a decimal which is \(0.5\), and then:\[4.8 + 0.5 = 5.3\].Therefore, the evaluated expression is \(5.3\).

Key Concepts

Substitution

Substitution is the initial step when evaluating expressions with variables. In this process, you replace the variables in the expression with their given numerical values. This step is crucial because it sets the foundation for the rest of the evaluation. When substituting, ensure to replace each variable accurately and carefully to avoid any errors in the subsequent steps. In our example, the variables are substituted as follows:

• Replace \(w\) with \(6\).
• Replace \(x\) with \(0.4\).
• Replace \(y\) with \(\frac{1}{2}\).

After substitution, the expression \(w-3x+y\) becomes \(6 - 3(0.4) + \frac{1}{2}\). This forms the foundation for solving the expression properly. Substitution is like drawing an accurate map before embarking on a journey—if it's incorrect, the directions you'll take will lead you astray.

Multiplication

Multiplication in expressions involves operating the factors involved to simplify the expression further. Once substitution is done, any coefficients of variables are to be multiplied by those substituted values. This is particularly important as it establishes the correct numbers to work with in the later steps. For our example, the expression \(6 - 3(0.4) + \frac{1}{2}\) requires:

• Multiplying \(3\) by \(0.4\) which equals \(1.2\).

This multiplication might seem simple, yet it's a pivotal step. It transforms our expression into \(6 - 1.2 + \frac{1}{2}\). Precision during multiplication ensures every subsequent calculation is based on accurate figures and leads to the correct final answer.

Subtraction

Subtraction is the operation of finding the difference between numbers. It follows multiplication in the operation hierarchy for this specific expression. In our expression \(6 - 1.2 + \frac{1}{2}\), subtraction lets us compute the real difference between the constant and the product from the multiplication. Doing it correctly means subtracting the smaller number from the larger one, thus:

• Subtract \(1.2\) from \(6\) to get \(4.8\).

The result \(4.8\) is the new expression's temporary result. Subtraction helps simplify the expression to a simpler form, reducing it to just one straightforward operation, which in this case, is to add the last component.

Addition

Addition is the operation that combines numbers to get their total or sum. In the given expression, addition is the final step that blends together all previous calculations into a single simplified result. With subtraction already done, the expression becomes \(4.8 + \frac{1}{2}\). To proceed:

• Convert \(\frac{1}{2}\) into a decimal \(0.5\).
• Add \(4.8\) and \(0.5\) to achieve \(5.3\).

Addition serves as the conclusive calculation in simplifying the expression. It bridges the gap between the partial result and the answer. Accurate addition ensures the entire sequence of operations culminates in the correct ultimate value. This simple process finalizes the expression evaluation, delivering the sought-after numerical outcome.