Problem 99 - Beginning and Intermediate Algebra

Question

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

Step-by-Step Solution

Answer

-5

Step 1: Identify the given values

The values provided are: \( x = 6 \), \( y = -4 \), and \( a = 3 \).

Step 2: Substitute the values into the expression

The expression is \(-2 y^{2} + 3 a^{2} \).Substituting the values, we get: \(-2 (-4)^{2} + 3 (3)^{2} \).

Step 3: Simplify the expression

First, calculate the squares: \( (-4)^{2} = 16 \) and \( (3)^{2} = 9 \).Substituting these, we get: \( -2 (16) + 3 (9) \)

Step 4: Perform the multiplications

Multiply the coefficients by the squared values: \( -2 \times 16 = -32 \) and \( 3 \times 9 = 27 \).So the expression now is: \( -32 + 27 \).

Step 5: Add the results

Add \( -32 \) and \( 27 \): \( -32 + 27 = -5 \).

Key Concepts

Substitution

Substitution is a powerful technique in algebra. It involves replacing variables in an expression with their given values. This is the first step when evaluating algebraic expressions.
In our exercise, we have the expression: \-2 y^{2} + 3 a^{2}. Using the given values \( y = -4 \) and \( a = 3 \), we substitute these values into the expression:
\[-2(-4)^{2} + 3 (3)^{2} \]
This turns our problem into a more manageable arithmetic expression that we can easily simplify.

Simplification

Simplification means making an expression easier to understand or solve without changing its value. It often involves combining like terms or performing arithmetic operations.
After substituting the values in our expression, we have: \-2 (-4)^{2} + 3 (3)^{2}.
We need to simplify further by first computing the squares:
\((-4)^{2} = 16 \) and \((3)^{2} = 9 \).
Substituting these back into the expression, we get:
\[-2 (16) + 3 (9) \].
This new expression is simpler to handle and is set for the next steps.

Order of Operations

The order of operations is a set of rules that determines the sequence in which mathematical operations should be performed to ensure accurate results.
This is often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right).
In our current expression\(-2 (16) + 3 (9)\), we need to follow these rules:
- First, handle the exponents (which are already done here).
- Next, perform any multiplications or divisions from left to right:
\(-2 * 16 = -32 \) and \(3 * 9 = 27 \).
- Finally, perform the addition (or subtraction in this case):
\(-32 + 27 \).

Evaluating Expressions

Evaluating expressions means calculating the value of the expression after performing all the operations.
In our problem, after simplifying and applying the order of operations, we end with the expression:
\[-32 + 27 \].
Adding these together, we get:
\(-32 + 27 = -5 \).
Thus, the value of the expression \-2 y^{2} + 3 a^{2}\, when \( y = -4 \) and \( a = 3 \), is -5.