Chapter 11: Q. 47 (page 773)

Solve the system. Use any method you wish.
$\{\begin{cases} x^{2} - 3 x y + 2 y^{2} = 0 \\ x^{2} + x y = 6 \end{cases}$

Short Answer

Expert verified

The solution of system of equations are $\{- \sqrt{3}, - \sqrt{3}\}, \{\sqrt{3}, \sqrt{3}\}, \{- 2, - 1\}, \{2, 1\}$ .

Step by step solution

Step 1. Given information.

Consider the given question,

$\{\begin{cases} x^{2} - 3 x y + 2 y^{2} = 0 . . . . . . (i) \\ x^{2} + x y = 6 . . . . . . (i i) \end{cases}$

Subtract equation (i) and (ii),

$(x^{2} - 3 x y + 2 y^{2}) - (x^{2} + x y) = 0 - 6 x^{2} - 3 x y + 2 y^{2} - x^{2} - x y = - 6 - 2 x y + y^{2} = - 3$

Step 2. Write in terms of x.

Writing in terms of x,

$2 x y = 3 + y^{2} x = \frac{3 + y^{2}}{2 y} . . . . . . (i i i)$

Substitute the value of xin equation (ii),

${(\frac{3 + y^{2}}{2 y})}^{2} + (\frac{3 + y^{2}}{2 y}) y = 6 \frac{9 + y^{4} + 6 y^{2}}{4 y^{2}} + \frac{3 + y^{2}}{2} = 6 (5 y^{2} + 3) (y^{2} - 3) = 0 y = - \sqrt{3}, \sqrt{3}, 1, - 1$

Step 3. Substitute y=-1,1 in equation (iii) and solve the equation.

Substitute $y = 1$ in equation (iii),

$x = \frac{3 + 1^{2}}{2 (1)} x = \frac{4}{2} x = 2$

Substitute $y = - 1$ in equation (iii),

$x = \frac{3 + {(- 1)}^{2}}{2 (- 1)} x = \frac{4}{- 2} x = - 2$

Step 4. Substitute y=-3,3 in equation (iii) and solve the equation.

Substitute $y = \sqrt{3}$ in equation (iii),

$x = \frac{3 + {(\sqrt{3})}^{2}}{2 \sqrt{3}} x = \frac{12}{2 \sqrt{3}} x = \sqrt{3}$

Substitute $y = - \sqrt{3}$ in equation (iii),

$x = \frac{3 + {(- \sqrt{3})}^{2}}{2 (- \sqrt{3})} x = \frac{12}{- 2 \sqrt{3}} x = - \sqrt{3}$

Therefore, the solution sets are $\{- \sqrt{3}, - \sqrt{3}\}, \{\sqrt{3}, \sqrt{3}\}, \{- 2, - 1\}, \{2, 1\}$ .

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Solve the system. Use any method you wish.
$\{\begin{cases} x^{2} - 3 x y + 2 y^{2} = 0 \\ x^{2} + x y = 6 \end{cases}$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Write in terms of x.

Step 3. Substitute y=-1,1 in equation (iii) and solve the equation.

Step 4. Substitute y=-3,3 in equation (iii) and solve the equation.

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Solve the system. Use any method you wish.x2-3xy+2y2=0x2+xy=6

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Write in terms of x.

Step 3. Substitute y=-1,1 in equation (iii) and solve the equation.

Step 4. Substitute y=-3,3 in equation (iii) and solve the equation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Geometry

Applied Mathematics

Discrete Mathematics

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Statistics

Calculus

Study anywhere. Anytime. Across all devices.

Solve the system. Use any method you wish.
$\{\begin{cases} x^{2} - 3 x y + 2 y^{2} = 0 \\ x^{2} + x y = 6 \end{cases}$