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Solve the following system of equations and show all work. y = x2 + 3 y = x + 5

Giving 20 points, please help.

Respuesta :

Luv2Teach

Answer:

(-1, 4) and (2, 7)

Step-by-step explanation:

If both of those equations equal y, the we can set them equal to each other and solve for x.  There will be 2 solutions because this is a second degree polynomial.

[tex]x^2+3=x+5[/tex]

Get everything on one side of the equals sign, set it equal to 0, then factor to solve for x.

[tex]x^2-x-2=0[/tex]

If you plug these values into the quadratic formula you will get x values of -1 and 2.  We first plug -1 into either one of the original equations to solve for y:

y = (-1) + 5 so

y = 4 and the resulting coordinate is (-1, 4)

Plug in 2 into the same equation to find y:

y = 2 + 5 so

y = 7

facundo3141592

We will see that the solutions of the system of equations are (-2, 3) and (1, 6).

Solving the system of equations:

So we have the system:

y = x^2 + 3

y = x + 5

We can see that the variable y is already isolated in both equations, then we can write:

x^2 + 3 = y = x + 5

x^2 + 3 = x + 5

Now we can solve the above equation for x, we will get:

x^2 + 3 - x - 5 = 0

x^2 - x - 2 = 0

This is a quadratic equation, the solutions are given by:

[tex]x = \frac{-1 \pm \sqrt{1^2 - 4*1*(-2)} }{2*1} \\\\x = \frac{-1 \pm 3}{2}[/tex]

Then the two solutions are:

  • x = (-1 - 3)/2 = -2
  • x = (-1 + 3)/2 = 1

To get the y-values we just replace these in one of the equations, we will get:

for x = -2

y = x + 5 = -2 + 5 = 3

Then the point (-2, 3) is a solution of the system.

for x = 1

y = x + 5 = 1 + 5 = 6

So the point (1, 6) is a solution of the system.

If you want to learn more about systems of equations, you can read:

https://brainly.com/question/13476446