The graph shows the equation y=x+2

Question

in red and the equation y=x2
in blue.





Which of the following shows the solution(s) for this system of equations?

(1 point)
Responses

(−1,1)
and (2,4)
open paren negative 1 comma 1 close paren and open paren 2 comma 4 close paren

(0,0)
and (0,2)
open paren 0 comma 0 close paren and open paren 0 comma 2 close paren

There are no solutions
There are no solutions

(4,2)

Answer 1

To find the solution(s) for the system of equations given by \(y = x + 2\) (the red line) and \(y = x^2\) (the blue parabola), we need to find the points where the two equations intersect.

1. Set the equations equal to each other: \[ x + 2 = x^2 \]

2. Rearranging gives us: \[ x^2 - x - 2 = 0 \]

3. Factoring the quadratic: \[ (x - 2)(x + 1) = 0 \]

4. Setting each factor to zero gives us the solutions: \[ x - 2 = 0 \quad \Rightarrow \quad x = 2 \] \[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

5. Now, we can find the corresponding \(y\) values for both \(x\) values using either original equation. Let's use \(y = x + 2\):

   • For \(x = 2\): \[ y = 2 + 2 = 4 \quad \Rightarrow \quad (2, 4) \]

   • For \(x = -1\): \[ y = -1 + 2 = 1 \quad \Rightarrow \quad (-1, 1) \]

Thus, the solutions for the system of equations are \((-1, 1)\) and \((2, 4)\).

The correct response is: (−1,1) and (2,4).