Solve x+y = 12 and 2xsquare + 3ysquare=7xy?

Answer 1

restate the first equation in terms of y:

y = 12 - x

Now substitute this in for y in the second equation:

2x² + 3(12 - x)² = 7x(12 - x)

Now, multiply it out and solve for x:

2x² + 3(144 - 24x + x²) = 84x - 7x²
2x² + 3x² + 7x² - 72x - 84x + 432 = 0
12x² - 156x + 432 = 0
12(x² - 13x + 36) = 0
x² - 13x + 36 = 0
(x - 9)(x - 4) = 0

x = 9 or 4

So, {x,y} = {9,3} or {4,8}

Answer 2

x + y = 12 and 2x² + 3y² = 7xy

Using the first equation, if we subtract y from both sides...
x = 12 - y

Putting our new found x value (12 - y) into the x variable in the other equation gives us...
2 * (12 - y)² + 3y² = 7(12 - y) * y
2(y² - 24y + 144) + 3y² = y(84 - 7y)
2y² - 48y + 288 + 3y² = 84y - 7y²
5y² - 48y + 288 = 84y - 7y²
12y² - 132y + 288 = 0 --- Divide both sides by 12...
y² - 11y + 24 = 0 / 12 = 0
(y - 8)(y - 3) = 0
y = 8 or y = 3

If y = 8, then x = 4. If y = 3, then x = 9. We plug both sets of numbers into the 2nd original equation to see if they work.
2(4²) + 3(8²) = 7(4)(8)
2(16) + 3(64) = 224
32 + 192 = 224
224 = 224 --- This means that x = 4 and y = 8 checks out. We should still try the other set of numbers:
2(9²) + 3(3²) = 7(9)(3)
2(81) + 3(9) = 189
162 + 27 = 189
189 = 189 --- This means that x = 9 and y = 3 also works.

Answer: (4, 8) and (9, 3) are both solutions to these equations; where the form of the answer is (x, y).

Answer 3

On the first equation, subtract x on both sides:
y=-x+12

Plug the equation into the y values:
2x^2+3(-x+12)^2=7x(-x+12)
2x^2+3(x^2-24x+144)=-7x^2+84x
2x^2+3x^2-72x+432=-7x^2+84x
5x^2-72x+432=-7x^2+84x
12x^2-156x+432=0
12(x^2-13x+36)=0
(x-4)(x-9)=0
x=4 and 9
4+y=12
9+y=12
y=8 and 3
The solution sets are (4,8) and (9,3)

Answer 4

x+y = 12
2xsquare + 3ysquare=7xy
2x^2-7xy+3y^2=0
(2x-y)(x-3y)=0
X=12-Y
(2(12-y)-y)(12-y-3y)=0
(24-3y)(12-4y)=0
24-3y=0
3y=24
y=8
x=12-8=4

1 root is (4,8)

12-4y=0
4y=12
y=3
x=12-3=9

the roots are (4,8), (9,3)

Answer 5

x+y=12
(x-3y)(2x-y)=0

x=12-y
(12-y-3y)(2(12-y)-y)=0
(-4y+12)(-2y+24)=0

y=3 and 12
x=9 and 0