Question

X-6 =-1/6y^2 and 2x+y=6. Solve the linear quadratic equation

Answer 1

`(4.5,-3)\textrm{ and }(0,6)`

Explanation:

Given:

`x-6=-(1)/(6)y^(2)`

`2x+y=6`

Express
`x` in terms of
`y` and then plug in first equation to solve for
`y`.

`2x+y=6\\2x=6-y\\x=(6-y)/(2)`

Now,

`x-6=-(1)/(6)y^(2)\\(6-y)/(2)-6=-(1)/(6)y^(2)\\(1)/(6)y^(2)+(6-y)/(2)-6=0`.

Multiply 6 on both sides.

`6((1)/(6)y^(2)+(6-y)/(2)-6)=6(0)\\y^(2)+3(6-y)-36=0\\y^(2)+18-3y-36=0\\y^(2)-3y-18=0\\(y-6)(y+3)=0\\y=6\textrm{ or }y=-3`

Now, for
`y=-3,x=(6-(-3))/(2)=(6+3)/(2)=(9)/(2)=4.5`

For,
`y=6, x=(6-6)/(2)=0`.

Therefore, the solutions for the system of equations are
`(4.5,-3)\textrm{ and }(0,6)`