Problem 81 - Elementary and Intermediate Algebra

Question

Evaluate each expression, for \(x=3, y=-2,\) and \(z=-4\) See Example 10. $$ 3 y^{2}-6 y-4 $$

Step-by-Step Solution

Answer

The expression evaluates to 20.

Step 1: Substitute the values

Substitute the given values of the variables into the expression. Here, substitute 3 for \(x\), \(-2\) for \(y\), and \(-4\) for \(z\). The expression is \(3y^{2} - 6y - 4\), so we substitute \(y = -2\):\[3(-2)^{2} - 6(-2) - 4\]

Step 2: Calculate the square and multiplication

First, calculate \((-2)^{2}\) which is \(4\). Then, multiply it by 3 to find the first term:\[3 \times 4 = 12\]

Step 3: Compute the second term

Calculate \(-6 \times (-2)\), which becomes positive due to the multiplication of two negative numbers:\[6 \times 2 = 12\]

Step 4: Add across the expression

Combine all the values from the calculations, remembering the constant term:\[12 + 12 - 4 = 20\]

Step 5: Final result

From the previous steps, the expression evaluates to 20. Therefore, the solution to the expression \(3y^{2} - 6y - 4\) when \(x=3\), \(y=-2\), and \(z=-4\) is 20.

Key Concepts

Substitution Method

When dealing with algebraic expressions, the substitution method involves replacing variables with their specific numeric values. This technique helps you convert variables into simple numbers, making the expression easier to evaluate. For example, in the given problem, we are asked to evaluate the expression \(3y^{2} - 6y - 4\) when \(y = -2\). By directly substituting \(-2\) for \(y\), the expression becomes:

• \(3(-2)^{2} - 6(-2) - 4\)

This step is crucial as it transforms an abstract mathematical concept into something more tangible.
Always double-check to ensure each variable is replaced correctly, as errors here can lead to incorrect answers.
Replacements should be consistent to avoid confusion.

Order of Operations

The order in which you perform mathematical operations is significant to get the correct result. This is known as the order of operations, often abbreviated as PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication
• Division
• Addition
• Subtraction

Say we have the expression \(3(-2)^{2} - 6(-2) - 4\).
Start with the exponents by evaluating the \((-2)^{2}\), resulting in \(4\). Then, multiply \(3\) by \(4\) to get \(12\).
Next, perform multiplication: \(-6\) times \(-2\) equals \(12\) because the product of two negatives is positive.
Finally, proceed with addition and subtraction: combine \(12 + 12 - 4\) to achieve the final result of \(20\).
Adhering to this order ensures that each step in solving the math problem proceeds logically and correctly.

Squared Numbers

When a number is squared, it is multiplied by itself. In mathematical notation, the squaring of a number \(x\) is written as \(x^{2}\). For instance, in the expression \(3y^{2} - 6y - 4\), the term \(y^{2}\) indicates that you should square the value of \(y\) first.
Here, since \(y = -2\), squaring it results in:

• \((-2)^{2} = 4\)

Notice that regardless of whether \(y\) is negative, the square of \(-2\) becomes positive. This occurs because multiplying a negative number by itself results in a positive number.
Understanding squared numbers is important as squaring alters the original value significantly, often forming the basis for more advanced calculations. Whenever you square a number, ensure that you follow through on the multiplication as accurately as possible to avoid computation errors.