Question

x^2+xy+y^2=7 and x^2-xy+y^2=3 solve it​

Answer 1

Required Answer:-

Given Information:

• x² + xy + y² = 7
• x² - xy + y² = 3

To find:

• The value of x and,
• The value of y.

Answer:

There are four solution to this equation. They are as follows..

1. x = 2 and y = 1
2. x = -2 and y = -1
3. x = -1 and y = -2
4. x = 1 and y = 2

Solution:

Given,

➡ x² + xy + y² = 7 ....(i)

➡ x² - xy + y² = 3 .....(ii)

On adding equations (i) and (ii), we get,

➡ 2(x² + y²) = 7 + 3

➡ 2(x² + y²) = 10

➡ x² + y² = 5

Substituting the value of x² + y² in equation (i), we get,

➡ 5 + xy = 7

➡ xy = 7 - 5

➡ xy = 2 (Remember this)

Now,

➡ x² + y² = 5

Adding 2xy on both sides, we get,

➡ x² + y² + 2xy = 5 + 2xy

➡ (x + y)² = 5 + 2 × 2

➡ (x + y)² = 9

➡ (x + y) = √9

➡ (x + y) = ±3

Again,

➡ x² + y² = 5

Subtracting 2xy from both sides, we get,

➡ x² + y² - 2xy = 5 - 2xy

➡ (x - y)² = 5 - 2 × 2

➡ (x - y)² = 5 - 4

➡ (x - y)² = 1

➡ (x - y) = √1

➡ (x - y) = ±1

So, when (x + y) = 3 and (x - y) = 1

➡ (x + y) + (x - y) = 3 + 1

➡ 2x = 4

➡ x = 2

Now,

y = 3 - x = 1

So, x = 2 and y = 1

Now, when (x + y) = -3 and (x - y) = -1

➡ (x + y) + (x - y) = -4

➡ 2x = -4

➡ x = -2

Now,

y = -3 - x

= -3 - (-2)

= -3 + 2

= -1

So, x = -2 and y = -1

Now, when (x + y) = -3 and (x - y) = 1

➡ (x + y) + (x - y) = -2

➡ 2x = -2

➡ x = -1

So,

y = -3 - x

= -3 - (-1)

= -3 + 1

= -2

So, x = -1 and y = -2

Now, when (x + y) = 3 and (x - y) = -1

➡ (x + y) + (x - y) = 3 - 1

➡ 2x = 2

➡ x = 1

So,

y = 3 - x

= 2

So, x = 1 and y = 2

So, there are four solutions to this equation. They are as follows.

1. x = 2 and y = 1
2. x = -2 and y = -1
3. x = -1 and y = -2
4. x = 1 and y = 2

Answer 2

Answer:

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