Problem 43 - Algebra 1

Question

Evaluate the expression. $$\frac{a^{2}-b^{2}}{2 c^{2}}+9 \text { when } a=-3, b=5, \text { and } c=-2$$

Step-by-Step Solution

Answer

The evaluated expression is 7.

Step 1: Substitution

The first step is to substitute the given values into the original equation. \(a = -3\), \(b = 5\), and \(c = -2\). Therefore, the equation becomes: \[\frac{(-3)^{2} - 5^{2}}{2 (-2)^{2}} + 9\]

Step 2: Simplification

Simplify the equation using the order of operations. First, simplify the terms in the numerator and the denominator separately. The equation becomes: \[\frac{9 - 25}{2 * 4} + 9\]Then, simplify the fraction:\[\frac{-16}{8} + 9\]The equation now becomes \[-2 + 9\]

Step 3: Final Calculation

Lastly, complete the arithmetic operation. \[-2 + 9 = 7.\]

Key Concepts

Substitution Method

The substitution method is a foundational mathematical approach used to evaluate expressions by replacing variables with their given numerical values. When faced with an expression that includes variables such as a, b, and c, you can only start to simplify it after substituting these placeholders with the real numbers you’ve been provided.

For example, consider the expression \(\frac{a^{2}-b^{2}}{2 c^{2}}+9\) when a=-3, b=5, and c=-2. To apply the substitution method, replace each variable with its corresponding value: \(\frac{(-3)^{2}-5^{2}}{2(-2)^{2}}+9\). Carefully swapping out each variable for its actual value is crucial to avoid mistakes early on in the problem-solving process.

Importance of Accuracy

Accuracy in substitution is vital. A single error in replacement can lead to an entirely incorrect result. Double-check your substitution before moving on to ensure that you're working with the correct numerical expression.

Order of Operations

Once you’ve completed the substitution, it’s time to simplify the expression. Simplification requires observing the order of operations, often remembered by the acronym PEMDAS, which stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

In our example, start by addressing the exponents and the operations within parentheses: \( (-3)^{2}-5^{2} \). Upon computing the squares, you proceed with subtraction inside the parentheses, followed by the division \(\frac{-16}{2(-2)^{2}}\). It is important to tackle multiplication or division as it appears from left to right in your expression, ensuring all such operations are addressed before moving to addition and subtraction. Ultimately, this leads you to the final, simplified expression \( -2 + 9 \), which is ready for the last step: addition.

Common Missteps

Students often miscalculate by ignoring the left-to-right rule for multiplication and division or by adding before subtracting. Remind yourself of the order of operations to avoid these common errors.

Simplifying Equations

Simplification makes complex equations more manageable and sets the stage for solving or evaluating them. In our exercise, simplification occurs after substitution and observing the order of operations. Begin by reducing fractions whenever possible and by combining like terms.

In the expression \(\frac{-16}{8} + 9\), simplification means dividing -16 by 8 to get -2, and then adding this result to 9. Simplifying the equation step by step prevents overwhelm and minimizes the risk of errors. When simplifying, always ensure that you have considered all reductions and combinations before claiming that your equation is in its simplest form.

Check Your Work

After each step, take a moment to review your calculations to confirm that you haven't missed a simpler form. This habit can save time in the long run and improve your accuracy when solving or evaluating equations. Simplification is more than a mechanical process; it is an exercise in attentiveness and thoroughness.