3xy-ysquare=2 and 2x -3y=-4
To solve the system of equations:
1. Rearrange the second equation to isolate one variable:
2x - 3y = -4
2x = 3y - 4
x = (3y - 4)/2
2. Substitute the expression for x in the first equation:
3xy - y^2 = 2
3(3y - 4)/2 * y - y^2 = 2
(9y^2 - 12y)/2 - y^2 = 2
9y^2 - 12y - 2y^2 = 4
3. Combine like terms:
7y^2 - 12y - 4 = 0
4. Solve this quadratic equation using factoring, completing the square, or the quadratic formula. In this case, we will use the quadratic formula:
y = (-b ± sqrt(b^2 - 4ac)) / 14
y = (-(-12) ± sqrt((-12)^2 - 4 * 7 * (-4))) / 14
y = (12 ± sqrt(144 + 112)) / 14
y = (12 ± sqrt(256)) / 14
y = (12 ± 16) / 14
5. Evaluate the possible values of y:
For y = (12 + 16) / 14, we get y ≈ 2
For y = (12 - 16) / 14, we get y ≈ -0.2857
6. Substitute the values of y back into the second equation to find the corresponding x values:
For y = 2:
2x - 3(2) = -4
2x - 6 = -4
2x = 2
x = 1
For y ≈ -0.2857:
2x - 3(-0.2857) = -4
2x + 0.8571 = -4
2x = -4.8571
x ≈ -2.4286
7. The solution to the system of equations is x = 1 and y = 2, or x ≈ -2.4286 and y ≈ -0.2857.
To solve this system of equations, we can use the method of substitution.
First, let's solve one equation for one of the variables. Let's solve the second equation, 2x - 3y = -4, for x:
2x - 3y = -4
2x = 3y - 4
x = (3y - 4)/2
Now that we have an expression for x in terms of y, we can substitute it into the first equation, 3xy - y^2 = 2:
3((3y - 4)/2)y - y^2 = 2
(9y^2 - 12y)y/2 - y^2 = 2
(9y^3 - 12y^2)/2 - y^2 = 2
(9y^3 - 12y^2)/2 = y^2 + 2
9y^3 - 12y^2 = 2(y^2 + 2)
9y^3 - 12y^2 = 2y^2 + 4
9y^3 - 14y^2 - 4 = 0
Now, we have a cubic equation in terms of y. We can solve this equation using algebraic methods or numerical methods, such as factoring or using a graphing calculator. Once we find the solutions for y, we can substitute them back into either of the original equations to find the corresponding values of x.
Please note that the solution to this system of equations may not have nice, integer solutions, depending on the values of y obtained.
3xy-y^2=2
2x-3y=-4
from the second, x=-2+1.5y
putting that into the first
3(-2+1.5y)y-y^2=2
-6y+4.5y^2-y^2=2
3.5y^2-6y-2=0
y^2-21y-7=0 check my work
quadratic equation..
y=(21+-sqrt(21^2+28)/2
y=(21+-21.66)/2
y- 21.33, or y=-.33
then calculate x for each.
CHECK ALL THIS, I did it in my head.