Solve the following system of equations and show all work: y = 2x², y = –3x − 1.

Equations are the backbone of mathematics. There can be many types of equations like quadratic equations, linear equations, and many others.

Answer: The solution of the system of equations y = 2x², y = –3x − 1 is x =  = -1, -1/2.

Let's understand the solution in detail.

Explanation: 

In this problem, we use the method of substitution to solve the problem

Given equations:

y = 2x² --------- (1)

y = –3x − 1 --------- (2)

Now, we substitute the value of (1) in (2)

⇒ 2x² = -3x - 1

⇒ 2x² + 3x + 1 = 0

Now, we use the quadratic formula to solve the equation above.

⇒ x = [-3 ± √{3² - 4(2)(1)}] / 2(2)

⇒ x = [-3 ± 1] / 4

⇒ x = -1, -1/2

Hence, the solution of the system of equations y = 2x², y = –3x − 1 is x =  = -1, -1/2.