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Chapter 0: Problem 99

Find the value of each expression if \(x=2\) and \(y=-1\). \(x^{2}+y^{2}\)

Short Answer

Expert verified
The value of the expression is 5.

Step by step solution

01

Identify the expression

The given expression to evaluate is \[ x^{2} + y^{2} \]
02

Substitute the given values

Replace \(x\) with 2 and \(y\) with -1 in the expression. So, it becomes: \[ (2)^{2} + (-1)^{2} \]
03

Evaluate the squares

Calculate the square of each substituted value: \[ 2^{2} = 4 \] \[ (-1)^{2} = 1 \]
04

Add the results

Add the results of the squared values: \[ 4 + 1 = 5 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Algebraic Substitution
Algebraic substitution is a straightforward technique where you replace variables with given values to simplify the expression. In this exercise, you are given the values of two variables: \(x = 2\) and \(y = -1\).
Substitution helps to make complex expressions easier to solve.
For instance, the original expression \(x^2 + y^2\) becomes much simpler when you substitute the values:
  • Replace \(x\) with 2, so \(x^2\) becomes \((2)^2\).
  • Replace \(y\) with -1, so \(y^2\) becomes \((-1)^2\).
After substitution, the expression looks like this: \((2)^2 + (-1)^2\).
This is much easier to handle and useful for breaking down even more complex algebraic expressions.
Square of a Number
Squaring a number means multiplying the number by itself. This concept is important as it often appears in many areas of math.
In our exercise, after substituting the values, the expression becomes \((2)^2 + (-1)^2\).
Let's break it down:
  • For \((2)^2\): Multiply 2 by 2. Thus, \(2 \times 2 = 4\).

  • For \((-1)^2\): Multiply -1 by -1. Thus, \((-1) \times (-1) = 1\).
It's important to note that squaring any real number, whether positive or negative, always gives a non-negative result.
This is because the product of two positive numbers is positive, and the product of two negative numbers is also positive.
Basic Arithmetic Operations
Basic arithmetic operations include addition, subtraction, multiplication, and division. In this exercise, after calculating the squares of the numbers, you need to add the results.
This is a simple operation where you combine values to get a sum.
Our expression after substitution and squaring looks like this: \(4 + 1\).
    To add 4 and 1, simply combine them: 4 + 1 = 5.
Understanding basic arithmetic operations is fundamental and frequently used in solving various math problems.
Once you master it, more complex equations become much easier to tackle.

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Most popular questions from this chapter

Expressions that occur in calculus are given. Factor each expression. Express your answer so that only positive exponents occur. $$\left(x^{2}+4\right)^{4 / 3}+x \cdot \frac{4}{3}\left(x^{2}+4\right)^{1 / 3} \cdot 2 x$$

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{\sqrt{2}}{\sqrt{7}+2}$$

Rationalize the denominator of each expression. Assume that all variables are positive when they appear. $$\frac{-3}{\sqrt{5}+4}$$

Simplify each expression. $$-25^{-1 / 2}$$

Simplify each expression. Assume that all variables are positive when they appear. $$\sqrt{15 x^{2}} \sqrt{5 x}$$
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