To solve the system of equations given by:
1. y = x^2 + 3 2. y = x + 5
we can set the two equations equal to each other since they both equal y:
x^2 + 3 = x + 5
Next, we’ll rearrange the equation to form a standard polynomial equation:
x^2 - x + 3 - 5 = 0
x^2 - x - 2 = 0
Now we can factor the quadratic equation:
(x - 2)(x + 1) = 0
This gives us two possible solutions for x:
x - 2 = 0 or x + 1 = 0
Thus, we find:
x = 2 or x = -1
Now, we need to substitute both values of x back into one of the original equations to find the corresponding y values. We can use the second equation:
y = x + 5
For the first solution (x = 2):
y = 2 + 5 = 7
For the second solution (x = -1):
y = -1 + 5 = 4
So, the solutions for the system of equations are:
1. (2, 7) 2. (-1, 4)
In conclusion, we have found the points where the two equations intersect: (2, 7) and (-1, 4).