Question

Tom determines the system of equations below has two solutions, one of which is located at the vertex of the parabola.

Equation 1: (x – 3)2 = y – 4
Equation 2: y = -x + b
In order for this solution to be reasonable, which qualifications must be met?
b must equal 7 and a second solution to the system must be located at the point (2, 5).
b must equal 1 and a second solution to the system must be located at the point (4, 5).
b must equal 7 and a second solution to the system must be located at the point (1, 8).
b must equal 1 and a second solution to the system must be located at the point (3, 4).

Answer 1

Option A: b must equal 7 and a second solution to the system must be located at the point (2, 5)

Step-by-step explanation:

step 1

Find the vertex of the quadratic equation

The general equation of a vertical parabola in vertex form is where (h, k) is the vertex we have so , the vertex is the point (3,4)

step 2

Find out the value of b in the linear equation we know that if the vertex is a solution of the system of equations, then the vertex must satisfy both equations substitute the value of x and the value of y of the vertex in the linear equation for x=3, y=4.

step 3

Find out the second solution of the system of equations we have

-----> equation A

----> equation B

Solve the system of equations by graphing . Remember that the solutions are the intersection points both graphs . The second solution of the system of equations is (2,5)

Therefore , b must equal 7 and a second solution to the system must be located at the point (2, 5)

Answer 2

b must equal 7 and a second solution to the system must be located at (2, 5).

Explanation:

Rearranging the first equation:

y = (x - 3)^2 + 4

From this we see that the vertex is at the point (3,4).

So one solution of equation 2 is (3 ,4).

Substituting in equation 2:

4 = -3 + b

b = 7.

So equation 2 is y = - x + 7.

Now we check if (2, 5) is on this line:

5 = -2 + 7 = 5 , therefore (2, 5) is on this line.

Verifying if (2, 5) is also on y = (x - 3)^2 + 4:

5 = (2 - 3)^2 + 4 = 1 + 4 = 5

- so it is. and a second solution to the system is (2, 5).