Problem 100 - Beginning and Intermediate Algebra

Question

Evaluate each expression for \(x=6, y=-4,\) and \(a=3\) \(5 x^{2}-4 y^{2}\)

Step-by-Step Solution

Answer

116

Step 1 - Substitute the values

Start by substituting the given values of the variables into the expression. So, for the expression \(5x^2 - 4y^2\), substitute \(x = 6\) and \(y = -4\). This gives: \[5(6)^2 - 4(-4)^2\]

Step 2 - Calculate the squares

Calculate the squares of the substituted values: \(6^2 = 36\) and \((-4)^2 = 16\). Substituting these values back into the expression gives: \[5 \times 36 - 4 \times 16\]

Step 3 - Multiply the constants

Next, multiply the squared terms by their respective coefficients: \(5 \times 36 = 180\) and \(4 \times 16 = 64\). So, the expression now reads: \[180 - 64\]

Step 4 - Subtract to get the final result

Subtract the second term from the first term: \(180 - 64 = 116\).

Key Concepts

Substitution Method

The substitution method is a key concept in algebra. It's essentially about replacing a variable with its actual value.
When evaluating an algebraic expression, you first need the specific values of the variables involved.
For example, in the exercise, we replaced variable values into the expression: Given, \( x = 6 \) and \( y = -4 \), they are placed into the expression \( 5x^2 - 4y^2 \).
This leads to: \(5(6)^2 - 4(-4)^2\).
By substituting values, we convert a general expression into a numerical form, making it simpler to solve.

Squaring Numbers

Squaring numbers is simply multiplying a number by itself. This operation is essential when working with algebraic expressions that include squared variables.
In our example, we calculated the squares of 6 and -4:

• \(6^2 = 6 \times 6 = 36\)
• \((-4)^2 = (-4) \times (-4) = 16\)

Remember, squaring a negative number will always yield a positive result because the product of two negative numbers is positive. This explains why \( (-4) ^2 = 16 \) is a positive value.

Order of Operations

The order of operations is a set of rules that ensures mathematical expressions are evaluated consistently. It consists of:

• Parentheses
• Exponents (including squaring numbers)
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

In our problem, following this order is crucial:

• First, we handled exponents, squaring 6 and -4
• Next, we performed the multiplications \(5 \times 36\) and \(4 \times 16\)
• Finally, we did the subtraction \(180 - 64\) to get the final result of 116

Following the order of operations prevents mistakes and ensures you reach the correct solution.