%5E2-(x%2B2)%5E2%3D9y-(y-3)%5E2-(y%2B2)%5E2%3D5x.jpeg "(x-1)^2-(x+2)^2=9y (y-3)^2-(y+2)^2=5x")
Solving the System of Equations: (x-1)^2 - (x+2)^2 = 9y and (y-3)^2 - (y+2)^2 = 5x
This article will guide you through the process of solving the given system of equations:
(1) (x-1)^2 - (x+2)^2 = 9y (2) (y-3)^2 - (y+2)^2 = 5x
Step 1: Simplify the Equations
We can simplify both equations by expanding the squares and combining like terms.
For equation (1):
- (x-1)^2 = x^2 - 2x + 1
- (x+2)^2 = x^2 + 4x + 4
- Substituting these into the equation: (x^2 - 2x + 1) - (x^2 + 4x + 4) = 9y
- Simplifying: -6x - 3 = 9y
- Further simplification: -2x - 1 = 3y
For equation (2):
- (y-3)^2 = y^2 - 6y + 9
- (y+2)^2 = y^2 + 4y + 4
- Substituting these into the equation: (y^2 - 6y + 9) - (y^2 + 4y + 4) = 5x
- Simplifying: -10y + 5 = 5x
- Further simplification: -2y + 1 = x
Step 2: Express one variable in terms of the other
We can now express one variable in terms of the other using the simplified equations. From the simplified equation (2), we have:
x = -2y + 1
Step 3: Substitute and Solve for one variable
Substitute the expression for 'x' in terms of 'y' into the simplified equation (1):
-2(-2y + 1) - 1 = 3y 4y - 2 - 1 = 3y y = 3
Step 4: Substitute and Solve for the other variable
Now, substitute the value of y (y = 3) back into the equation x = -2y + 1:
x = -2(3) + 1 x = -6 + 1 x = -5
Solution
Therefore, the solution to the system of equations is x = -5 and y = 3.
Verification
We can verify this solution by substituting these values back into the original equations:
(1) (-5 - 1)^2 - (-5 + 2)^2 = 9(3) --> 36 - 9 = 27 --> True (2) (3 - 3)^2 - (3 + 2)^2 = 5(-5) --> -25 = -25 --> True
This verifies that our solution is correct.