Question

solve each system of equations

6x^2+3y^2-111x-y+128=0
3x-y=1

Answer 1

from the 2nd:
y = 3x - 1
then
6x^2+3y^2-111x-y+128=0
6x^2 + 3(3x - 1)^2 - 111x - (3x - 1) + 128 = 0
simplify, and solve the equation.
(comes apart nicely)

Answer 2

6x^2+3y^2-111x-y+128=0 , 3x-y=1


3 x - y = 1

Subtract 3x to both sides:

- y = 1 - 3 x

- y = - 3 x + 1

Multiply both sides by - 1

y = 3 x - 1

plug y = 3 x - 1 into 6 x² + 3 y² -111 x - y +128 = 0

6 x² + 3 ∙ ( 3 x - 1 )² -111 x - ( 3 x - 1 ) + 128 = 0

6 x² + 3 ∙ ( 9 x² - 6 x + 1 ) - 111 x - 3 x + 1 + 128 = 0

6 x² + 27 x² - 18 x + 3 - 111 x - 3 x + 1 + 128 = 0

33 x² -132 x + 132 = 0

Divide both sides by 33

x² - 4 x + 4 = 0

Solution of this equation:

x = 2


y = 3 x - 1 = 3 ∙ 2 - 1 = 6 - 1 = 5

Solution x = 2 , y = 5

( 2 , 5 )