Chapter 11: Q. 43 (page 773)

Solve the system. Use any method you wish.
$\{\begin{cases} \frac{5}{x^{2}} - \frac{2}{y^{2}} + 3 = 0 \\ \frac{3}{x^{2}} + \frac{1}{y^{2}} = 7 \end{cases}$

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information.

Consider the given question,

$\{\begin{cases} \frac{5}{x^{2}} - \frac{2}{y^{2}} + 3 = 0 . . . . . . (i) \\ \frac{3}{x^{2}} + \frac{1}{y^{2}} = 7 . . . . . . (i i) \end{cases}$

Multiply equation (i) by $x^{2} y^{2}$ ,

localid="1646926488110" $\frac{5 x^{2} y^{2}}{x^{2}} - \frac{2 x^{2} y^{2}}{y^{2}} + 3 x^{2} y^{2} = 0 5 y^{2} - 2 x^{2} = - 3 x^{2} y^{2} . . . . . . (i i i)$

Multiply equation (ii) by $x^{2} y^{2}$ ,

localid="1646926183062" role="math" $\frac{3 x^{2} y^{2}}{x^{2}} + \frac{x^{2} y^{2}}{y^{2}} = 7 x^{2} y^{2} 3 y^{2} + x^{2} = 7 x^{2} y^{2} . . . . . . (i v)$

Step 2. Multiply equation (iv) by 2.

Multiply equation (iv) by $2$ ,

$6 y^{2} + 2 x^{2} = 14 x^{2} y^{2} . . . . . . (v)$

Add equation (iii) and (v),

role="math" localid="1646926512030" $(5 y^{2} - 2 x^{2}) + (6 y^{2} + 2 x^{2}) = 14 x^{2} y^{2} - 3 x^{2} y^{2} 11 y^{2} = 11 x^{2} y^{2} x^{2} = 1 x = - 1, 1$

Step 3. Substitute the values of x in equation (iv).

Substitute $x = - 1$ in equation (iv),

localid="1646926643291" $3 y^{2} + {(- 1)}^{2} = 7 {(- 1)}^{2} y^{2} 3 y^{2} + 1 = 7 y^{2} 1 = 4 y^{2} y = - \frac{1}{2}, \frac{1}{2}$

Substitute $x = 1$ in equation (iv),

$3 y^{2} + 1^{2} = 7 {(1)}^{2} y^{2} 3 y^{2} + 1 = 7 y^{2} 1 = 4 y^{2} y = - \frac{1}{2}, \frac{1}{2}$

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

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Solve the system. Use any method you wish.
$\{\begin{cases} \frac{5}{x^{2}} - \frac{2}{y^{2}} + 3 = 0 \\ \frac{3}{x^{2}} + \frac{1}{y^{2}} = 7 \end{cases}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Multiply equation (iv) by 2.

Step 3. Substitute the values of x in equation (iv).

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Solve the system. Use any method you wish.5x2-2y2+3=03x2+1y2=7

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Multiply equation (iv) by 2.

Step 3. Substitute the values of x in equation (iv).

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Logic and Functions

Calculus

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Study anywhere. Anytime. Across all devices.

Solve the system. Use any method you wish.
$\{\begin{cases} \frac{5}{x^{2}} - \frac{2}{y^{2}} + 3 = 0 \\ \frac{3}{x^{2}} + \frac{1}{y^{2}} = 7 \end{cases}$