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What are the solutions to the simultaneous equations of x squared -xy -y squared equals -11 and 2x plus y equals 1 showing work?
Wiki User
∙ 7y ago
The two equations are:
- x² - xy - y² = -11
- 2x + y = 1
The easiest way is to eliminate one of the unknowns by rearranging (2) so that y can be expressed as function of x, substituting this into (1) which will give a quadratic involving only x which can then be solved. (Alternatively rearranging 2 to express x in terms of y and then substituting for x in (1) can be done, but this leads to fractions which are not so easy to work with.) Once the values of x have been found, equation (2) can be used to find the corresponding values of y.
Rearranging (2) to make y the subject gives y = 1 -2x
Substituting this for y in (1) gives:
x² - x(1 - 2x) - (1 - 2x)² = -11
→ x² - x + 2x² - (1 - 4x + 4x²) = -11
→ 3x² - x - 1 + 4x - 4x² = -11
→ -x² + 3x - 1 = -11
→ x² - 3x - 10 = 0
→ (x - 5)(x + 2) = 0
→ x = 5 or -2
Using (2) this gives:
x = 5 → y = 1 - 2×5 = -9
x = -2 → y = 1 - 2×-2 = 5
→ The solutions are the points (5, -9) and (-2, 5)
Wiki User
∙ 7y ago
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