how do i solve for x and y x+y=2 and 3x^2+4xy+y^2=6
You have to isolate one variable in the first equation first because its easier, then substitute it into the second one:
x=2-y
Now plug this in for x in the second equation:
3(2-y)^2+4(2-y)y+y^2=6
Now solve for y, then plug your answer back into:
x=2-y
3(2-y)(2-y)+8y-4y^2+y^2=6
3(4-4y+y^2)+8y-3y^2=6
12-12y+3y^2+8y-3y^2=6
-4y+12=6
-4y=6-12
y=-1.5
Now plug it in:
x=2-(-1.5)
x=3.5
Hope that helped! :)
my bad, the 1.5 should be positive! so y=1.5 and x=0.5
x + y = 2 Subtract x to both sides
x + y - x = 2 - x
y = 2 - x
3 x ^ 2 + 4 x y + y ^ 2 = 6
3 x ^ 2 + 4 x ( 2 - x ) + ( 2 - x ) ^ 2 = 6
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Remark :
( a - b ) ^ 2 = a ^ 2 - 2 a b + b ^ 2
So :
( 2 - x ) ^ 2 = 2 ^ 2 - 2 * 2 *x + x ^ 2
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3 x ^ 2 + 4 x * 2 + 4 x * ( - x ) + 2 ^ 2 - 2 * 2 * x + x ^ 2 = 6
3 x ^ 2 + 8 x - 4 x ^ 2 + 4 - 4 x + x ^ 2 = 6
3 x ^ 2 - 4 x ^ 2 + x ^ 2 + 8 x + 4 - 4 x = 6
- x ^ 2 + x ^ 2 + 8 x - 4 x + 4 = 6
4 x + 4 = 6 Subtract 4 to both sides
4 x + 4 - 4 = 6 - 4
4 x = 2 Divide both sides by 4
4 x / 4 = 2 / 4
x = 2 / ( 2 * 2 )
x = 1 / 2
y = 2 - x
y = 2 - 1 / 2
y = 4 / 2 - 1 / 2
y = 3 / 2
The solutions are :
x = 1 / 2 , y = 3 / 2
To solve for x and y in the given system of equations:
1. Start with the first equation: x + y = 2.
We can solve this equation for x or y by isolating one of the variables. Let's solve for y:
Subtract x from both sides:
x + y - x = 2 - x
Simplify:
y = 2 - x
2. Substitute the value of y in terms of x into the second equation:
3x^2 + 4xy + y^2 = 6
Replace y with 2 - x:
3x^2 + 4x(2 - x) + (2 - x)^2 = 6
Simplify and expand:
3x^2 + 8x - 4x^2 + 4 - 4x + x^2 = 6
Combine like terms:
3x^2 - 4x^2 + x^2 + 8x - 4x - 4 + 4 = 6
Simplify further:
0x^2 + 5x + 0 = 6
Now we have a quadratic equation: 5x = 6
3. Solve the quadratic equation to find the value(s) of x:
Write the equation in standard form:
5x - 6 = 0
To solve, you can use factoring, completing the square, or the quadratic formula. In this case, we will use factoring:
Factor the equation:
(x - 6/5)(5) = 0
Set each factor equal to zero:
x - 6/5 = 0
Solve for x:
x = 6/5
4. Substitute the value of x (x = 6/5) back into the first equation to find the corresponding value of y:
x + y = 2
Substituting x = 6/5:
6/5 + y = 2
Subtract 6/5 from both sides:
y = 2 - 6/5
Simplify:
y = 10/5 - 6/5
y = 4/5
So, the solution to the system of equations is x = 6/5 and y = 4/5.