Solve the Nonlinear equation in a)Substitution Methods b)Graphical Methods  1.         y = 2x2 ------------------①           2x – y + 4 = 0 ----------②  2.        x2 + y2 = 25 ------------①           2x – 3y = - 6   ------------②   3.        x2 – 4y2 = 200 ---------①           x + 2y = 100   ---------②

To solve the given nonlinear equations using substitution methods:

a) For equation 1: We have y = 2x^2 (equation ①) and 2x - y + 4 = 0 (equation ②).

Step 1: Solve equation ② for y: Rearrange equation ② to get y = 2x + 4.

Step 2: Substitute the value of y from step 1 into equation ①: Replace y with 2x + 4 in equation ①: 2x^2 = 2x + 4.

Step 3: Simplify and solve for x: Rearrange the equation to get 2x^2 - 2x - 4 = 0. Solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the values of x.

b) For equation 2: We have x^2 + y^2 = 25 (equation ①) and 2x - 3y = -6 (equation ②).

Step 1: Solve equation ② for x: Rearrange equation ② to get x = (3y - 6)/2.

Step 2: Substitute the value of x from step 1 into equation ①: Replace x with (3y - 6)/2 in equation ①: (3y - 6)^2/4 + y^2 = 25.

Step 3: Simplify and solve for y: Expand and rearrange the equation to get a quadratic equation in terms of y. Solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the values of y.

c) For equation 3: We have x^2 - 4y^2 = 200 (equation ①) and x + 2y = 100 (equation ②).

Step 1: Solve equation ② for x: Rearrange equation ② to get x = 100 - 2y.

Step 2: Substitute the value of x from step 1 into equation ①: Replace x with 100 - 2y in equation ①: (100 - 2y)^2 - 4y^2 = 200.

Step 3: Simplify and solve for y: Expand and rearrange the equation to get a quadratic equation in terms of y. Solve this quadratic equation using factoring, completing the square, or the quadratic formula to find the values of y.

Once you have found the values of x and y, substitute them back into the original equations to verify if they satisfy the equations.