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Chapter 2: Problem 9

For the following exercises, evaluate the algebraic expressions. If \(y=\frac{x+1}{2-x},\) evaluate \(y\) given \(x=5 i\).

Short Answer

Expert verified
\( y = \frac{-23}{29} + \frac{15i}{29} \) when \( x = 5i \).

Step by step solution

01

Identify the given expression and substitute the value of x

The algebraic expression given is \( y = \frac{x+1}{2-x} \). We need to evaluate \( y \) when \( x = 5i \). Substitute \( x = 5i \) into the expression to get: \[ y = \frac{5i + 1}{2 - 5i} \]
02

Simplify the expression by rationalizing the denominator

To simplify the expression, rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator, \( 2 + 5i \):\[ y = \frac{(5i + 1)(2 + 5i)}{(2 - 5i)(2 + 5i)} \]The denominator \((2 - 5i)(2 + 5i)\) becomes:\[ (2)^2 - (5i)^2 = 4 - (25i^2) = 4 - (-25) = 29 \]
03

Expand the numerator

Expand the numerator \((5i + 1)(2 + 5i)\):\[ = 5i \cdot 2 + 5i \cdot 5i + 1 \cdot 2 + 1 \cdot 5i \]This results in:\[ = 10i + 25i^2 + 2 + 5i \]Since \(i^2 = -1\), substitute it in:\[ = 10i - 25 + 2 + 5i \]Combine like terms:\[ = (-23) + 15i \]
04

Write the simplified expression

After simplifying the numerator and denominator, the expression becomes:\( y = \frac{-23 + 15i}{29} \).
05

Separate the real and imaginary parts

Separate into real and imaginary components:\[ y = \frac{-23}{29} + \frac{15i}{29} \]
06

Conclusion

Thus, the value of \( y \) when \( x = 5i \) is:\[ y = \frac{-23}{29} + \frac{15i}{29} \]

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Key Concepts

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

Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operation symbols that define a specific mathematical relationship or rule. In this exercise, our algebraic expression is given by \( y = \frac{x+1}{2-x} \). When working with algebraic expressions, it's crucial to remember that they can be simplified, evaluated by substituting variables with specific values, and manipulated using various algebraic techniques.
  • Variables: Represent unknown values and can change within expressions.
  • Constants: The numeric values within an expression that do not change.
  • Operations: Include addition, subtraction, multiplication, division, and sometimes exponents.
Once you substitute a value for a variable, like \( x = 5i \) in our case, the expression can be further simplified using the rules of complex numbers and algebra.
Imaginary Unit
The imaginary unit, denoted as \( i \), is a mathematical concept that allows for the existence of complex numbers. The defining feature of \( i \) is that \( i^2 = -1 \). This property is fundamental in extending numbers beyond the real number line to include complex numbers, which have both real and imaginary parts.
  • Complex Numbers: Written as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.
  • Uses of \( i \): Essential in many fields, including engineering and physics, to solve equations that have no real solutions.
In our exercise, by substituting \( x = 5i \), we engage with both real and imaginary aspects, requiring a careful approach to simplify the expression.
Rationalizing Denominators
Rationalizing the denominator involves eliminating any imaginary numbers from the denominator of a fraction. This is typically achieved by multiplying the numerator and the denominator by the conjugate of the denominator.For instance, in the exercise, the expression \( y = \frac{5i + 1}{2 - 5i} \) requires multiplication with the conjugate, \( 2 + 5i \). This procedure:
  • Cleans the denominator: \( (2 - 5i)(2 + 5i) = 4 + 25 = 29 \), converting it to a real number.
  • Maintains equality: Because we multiply both the top and bottom of the fraction by the same term, the value of the expression remains unchanged.
This step ensures that the expression is in a simplified and standard form, ready for further calculation.
Complex Conjugate
A complex conjugate of a complex number \( a + bi \) is \( a - bi \). The multiplication of a complex number by its conjugate results in a real number, as seen in the rationalization process.For the expression in the exercise, the denominator \( 2 - 5i \) has a conjugate \( 2 + 5i \). Multiplying by this conjugate simplifies the expression by eliminating the imaginary component from the denominator.
  • Real Result: \( (a+bi)(a-bi) = a^2 + b^2 \), which is always a real number.
  • Utility: Complex conjugates are useful not only in simplifying fractions but also in various calculations in mathematics and physics.
Understanding complex conjugates helps navigate through calculations that involve complex numbers, ensuring the results are in their simplest form.

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Most popular questions from this chapter

For the following exercises, input the left-hand side of the inequality as a \(Y 1\) graph in your graphing utility. Enter \(y 2=\) the right-hand side. Entering the absolute value of an expression is found in the MATH menu, Num, \(1: a b s(.\) Find the points of intersection, recall \(\left(2^{\text {nd }}\right.\) CALC 5 :intersection, \(1^{\text {st }}\) curve, enter, \(2^{\text {nd }}\) curve, enter, guess, enter). Copy a sketch of the graph and shade the \(x\) -axis for your solution set to the inequality. Write final answers in interval notation. $$ \frac{-1}{2}|x+2|<4 $$

For the following exercises, solve the inequality. Write your final answer in interval notation. $$ 3 x+2 \geq 7 x-1 $$

For the following exercises, describe all the \(x\) -values within or including a distance of the given values. Distance of 5 units from the number 7

For the following exercises, solve the rational exponent equation. Use factoring where necessary. $$ 2 x^{\frac{1}{2}}-x^{\frac{1}{4}}=0 $$

For the following exercises, graph both straight lines (left-hand side being y1 and right-hand side being \(y 2\) ) on the same axes. Find the point of intersection and solve the inequality by observing where it is true comparing the \(y\) -values of the lines. $$ 4 x+1<\frac{1}{2} x+3 $$
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