Evaluate the following expression for the given values:
[tex]$-5 x^2-2 y^2+1$[/tex] where [tex]$x=3$[/tex], [tex]$y=-4$[/tex]
Answer (2)
Substitute x = 3 and y = − 4 into the expression − 5 x 2 − 2 y 2 + 1 .
Calculate x 2 = 3 2 = 9 and y 2 = ( − 4 ) 2 = 16 .
Substitute the squared values: − 5 ( 9 ) − 2 ( 16 ) + 1 = − 45 − 32 + 1 .
Simplify to get the final answer: − 45 − 32 + 1 = − 76 . The final answer is − 76 .
Explanation
Understanding the Problem We are asked to evaluate the expression − 5 x 2 − 2 y 2 + 1 when x = 3 and y = − 4 . This involves substituting the given values of x and y into the expression and simplifying.
Substitution First, we substitute x = 3 and y = − 4 into the expression: − 5 ( 3 ) 2 − 2 ( − 4 ) 2 + 1
Calculating Squares Next, we calculate the squares of x and y :
x 2 = 3 2 = 9 y 2 = ( − 4 ) 2 = 16
Substituting Squares Now, we substitute these values back into the expression: − 5 ( 9 ) − 2 ( 16 ) + 1
Performing Multiplications Next, we perform the multiplications: − 5 ( 9 ) = − 45 − 2 ( 16 ) = − 32
Simplifying the Expression Finally, we substitute these results back into the expression and simplify: − 45 − 32 + 1 = − 77 + 1 = − 76
Final Answer Therefore, the value of the expression − 5 x 2 − 2 y 2 + 1 when x = 3 and y = − 4 is − 76 .
Examples
Understanding how to evaluate algebraic expressions is crucial in many real-world applications. For instance, in physics, you might use such expressions to calculate the potential energy of an object, where variables represent position or other physical quantities. In economics, you could model cost functions or revenue streams using similar algebraic structures. Evaluating these expressions allows you to make predictions and informed decisions based on the values of the variables involved. For example, the expression could represent the profit of a small business, where x is the number of items sold and y is the cost of production. By substituting different values for x and y, the business owner can estimate their profit under various scenarios.
The expression − 5 x 2 − 2 y 2 + 1 evaluates to − 76 when substituting x = 3 and y = − 4 . This is done by calculating the squares, performing the multiplications, and simplifying the resulting expression. Therefore, the final answer is − 76 .
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