Question

Solve the system of equations: x^2 + y^2 = 1 3x + y = -1

Answer 1

(0, - 1 ) , (-
`(3)/(5)`,
`(4)/(5)` )

Explanation:

given the 2 equations to solve simultaneously

x² + y² = 1 → (1)

3x + y = - 1 ( subtract 3x from both sides )

y = - 1 - 3x → (2)

substitute y = - 1 - 3x into (1)

x² + (- 1 - 3x)² = 1 ← expand parenthesis using FOIL

x² + 1 + 6x + 9x² = 1

10x² + 6x + 1 = 1 ( subtract 1 from both sides )

10x² + 6x = 0 ← factor out 2x from each term on the left side

2x(5x + 3) = 0 ← in factored form

equate each factor to zero and solve for x

2x = 0 ⇒ x = 0

5x + 3 = 0 ( subtract 3 from both sides )

5x = - 3 ( divide both sides by 5 )

x = -
`(3)/(5)`

substitute these values of x into (2) for corresponding y values

x = 0 : y = - 1 - 3(0) = - 1 - 0 = - 1 ⇒ (0, - 1 )

x = -
`(3)/(5)` : y = - 1 - 3(-
`(3)/(5)` ) = -
`(5)/(5)` +
`(9)/(5)` =
`(4)/(5)` ⇒ (-
`(3)/(5)`,
`(4)/(5)` )

solutions are (0, - 1 ) and (-
`(3)/(5)`,
`(4)/(5)` )