Problem 34 - Intermediate Algebra

Question

In the following exercises, evaluate the following expressions. When \(x=3, y=2\) \(6 x^{2}+3 x y-9 y^{2}\)

Step-by-Step Solution

Answer

36

Step 1: Substitute the Values

First, substitute the values of the variables into the expression. For this exercise, substitute \(x = 3\) and \(y = 2\) into the given expression: \[6 x^{2} + 3 x y - 9 y^{2}\]

Step 2: Calculate the Squared Terms

Next, calculate the squared terms. This means you need to find \( x^2 \) and \( y^2 \): \(3^2 = 9\) and \(2^2 = 4\).

Step 3: Multiply by the Coefficients

Multiply the squared terms calculated in Step 2 by their respective coefficients in the equation: \(6 \times 9 = 54\) for \(6 x^2\) and \(9 \times 4 = 36\) for \(9 y^2\).

Step 4: Calculate the Product Term

Calculate the product term, \(3 x y\), using the substituted values: \(3 \times 3 \times 2 = 18\).

Step 5: Combine the Results

Now combine all the calculated results from the steps above into the expression: \[54 + 18 - 36\].

Step 6: Simplify the Expression

Finally, simplify the combined results to find the value of the expression: \[54 + 18 - 36 = 36\].

Key Concepts

Substitution

Substitution is a fundamental concept in evaluating algebraic expressions. It involves replacing variables in an expression with their designated values. In our exercise, we replace every instance of the variables with their given values: we substitute \(x = 3\) and \(y = 2\). This transforms the original expression, which helps simplify it step-by-step.
To visualize, the substituted expression becomes: \[6(3)^2 + 3(3)(2) - 9(2)^2\].
Each variable now has a specific number, making the process of evaluating the expression straightforward.

Squared Terms

Squared terms appear frequently in algebra. Squaring a term means multiplying the term by itself. In our exercise, the squared terms are \(x^2\) and \(y^2\). After substitution, we calculate these as follows:

• \(x^2 = 3^2 = 9\)


• \(y^2 = 2^2 = 4\)

Squared terms help in determining parts of the expression’s values independently before combining them back together.
It’s crucial to square the terms accurately to ensure the overall expression simplifies correctly.

Coefficients

Coefficients are numbers placed in front of variables or terms in algebraic expressions. They indicate how many instances of the variable or term are present. In our example:

• The coefficient of \(x^2\) is 6.


• The coefficient of \(y^2\) is 9.


• The coefficient of \(xy\) is 3.

After calculating the squared terms, we multiply them by their respective coefficients. For example, \(6(9)\) for \(6x^2\) and \(9(4)\) for \(9y^2\).
This multiplication is crucial for accurate evaluation and simplification of the expression.

Simplifying Expressions

Simplifying expressions is about combining like terms and reducing an expression to its simplest form. Here, we combine results from our calculations:
After substitution and computing squared terms, our expression became \[6(9) + 3(3)(2) - 9(4)\].
We then calculated:

• \(6\times 9 = 54\)


• \(3\times 3\times 2 = 18\)


• \(9\times 4 = 36\)

Combine them all: \[54 + 18 - 36 = 36\] which is the simplified value of the expression.
Simplification makes the expression much easier to interpret and work with.