Q. 46 Solve the system. Use any method... [FREE SOLUTION]

91影视

Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 11: Q. 46 (page 773)

Solve the system. Use any method you wish.
$\{\begin{cases} \frac{1}{x^{4}} - \frac{1}{y^{4}} = 1 \\ \frac{1}{x^{4}} + \frac{1}{y^{4}} = 4 \end{cases}$

Short Answer

Expert verified

The solution of system of equations are $\{- {(\frac{2}{5})}^{\frac{1}{4}}, - {(\frac{2}{3})}^{\frac{1}{4}}\}, \{- {(\frac{2}{5})}^{\frac{1}{4}}, {(\frac{2}{3})}^{\frac{1}{4}}\}, \{{(\frac{2}{5})}^{\frac{1}{4}}, - {(\frac{2}{3})}^{\frac{1}{4}}\}, \{{(\frac{2}{5})}^{\frac{1}{4}}, {(\frac{2}{3})}^{\frac{1}{4}}\} .$

Step by step solution

Step 1. Given information.

Consider the given question,

$\{\begin{cases} \frac{1}{x^{4}} - \frac{1}{y^{4}} = 1 . . . . . . (i) \\ \frac{1}{x^{4}} + \frac{1}{y^{4}} = 4 . . . . . . (i i) \end{cases}$

Multiply equation (i) by $x^{4} y^{4}$ ,

$\frac{x^{4} y^{4}}{x^{4}} - \frac{x^{4} y^{4}}{y^{4}} = x^{4} y^{4} y^{4} - x^{4} = x^{4} y^{4} . . . . . . (i i i)$

Multiply equation (ii) by $x^{4} y^{4}$ ,

$\frac{x^{4} y^{4}}{x^{4}} + \frac{x^{4} y^{4}}{y^{4}} = 4 x^{4} y^{4} y^{4} + x^{4} = 4 x^{4} y^{4} . . . . . . (i v)$

Step 2. Add equations (iii) and (iv).

Add equations (iii) and (iv),

$(y^{4} - x^{4}) + (y^{4} + x^{4}) = x^{4} y^{4} + 4 x^{4} y^{4} 2 y^{4} = 5 x^{4} y^{4} x^{4} = \frac{2}{5} x = 卤 {(\frac{2}{5})}^{\frac{1}{4}}$

Substitute $x = {(\frac{2}{5})}^{\frac{1}{4}}$ in equation (iii),

role="math" localid="1646933679679" $y^{4} - {\{{(\frac{2}{5})}^{\frac{1}{4}}\}}^{4} = {\{{(\frac{2}{5})}^{\frac{1}{4}}\}}^{4} y^{4} y^{4} - \frac{2}{5} = \frac{2}{5} y^{4} y^{4} = \frac{2}{3} y = {(- \frac{2}{3})}^{\frac{1}{4}}, {(\frac{2}{3})}^{\frac{1}{4}}$

Step 3. Substitute x=-2514 in equation (iii) and solve the equation.

Substitute $x = - {(\frac{2}{5})}^{\frac{1}{4}}$ in equation (iii),

role="math" localid="1646933701331" $y^{4} - {\{- {(\frac{2}{5})}^{\frac{1}{4}}\}}^{4} = {\{- {(\frac{2}{5})}^{\frac{1}{4}}\}}^{4} y^{4} y^{4} - \frac{2}{5} = \frac{2}{5} y^{4} y^{4} = \frac{2}{3} y^{4} = {(- \frac{2}{3})}^{\frac{1}{4}}, {(\frac{2}{3})}^{\frac{1}{4}}$

Therefore, the solution sets are $\{- {(\frac{2}{5})}^{\frac{1}{4}}, - {(\frac{2}{3})}^{\frac{1}{4}}\}, \{- {(\frac{2}{5})}^{\frac{1}{4}}, {(\frac{2}{3})}^{\frac{1}{4}}\}, \{{(\frac{2}{5})}^{\frac{1}{4}}, - {(\frac{2}{3})}^{\frac{1}{4}}\}, \{{(\frac{2}{5})}^{\frac{1}{4}}, {(\frac{2}{3})}^{\frac{1}{4}}\}$ .

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91影视

Solve the system. Use any method you wish.
$\{\begin{cases} \frac{1}{x^{4}} - \frac{1}{y^{4}} = 1 \\ \frac{1}{x^{4}} + \frac{1}{y^{4}} = 4 \end{cases}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Add equations (iii) and (iv).

Step 3. Substitute x=-2514 in equation (iii) and solve the equation.

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91影视

Solve the system. Use any method you wish.1x4-1y4=11x4+1y4=4

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Add equations (iii) and (iv).

Step 3. Substitute x=-2514 in equation (iii) and solve the equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Applied Mathematics

Statistics

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Mechanics Maths

Study anywhere. Anytime. Across all devices.

Solve the system. Use any method you wish.
$\{\begin{cases} \frac{1}{x^{4}} - \frac{1}{y^{4}} = 1 \\ \frac{1}{x^{4}} + \frac{1}{y^{4}} = 4 \end{cases}$