Question

Tom determines that 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 Tom’s thinking to be correct, 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

SOLUTION

GIVEN

Tom determines that 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 Tom’s thinking to be correct, 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).

EVALUATION

Here it is given that

Tom determines that 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

Now vertex of parabola is (3,4)

So one solution is (3,4)

Now the (3,4) satisfies the equation y = - x + b

Which gives

4 = - 3 + b

⇒ b = 7

Now Putting the value of B we get

y = - x + 7

From Equation 1 we get

$\sf{ {(x - 3)}^{2} = - x + 7 - 4}$

$\sf{ \implies \: {x}^{2} - 6x + 9 - 7 + x + 4 = 0}$

$\sf{ \implies \: {x}^{2} - 5x + 6= 0}$

$\sf{ \implies \: {x}^{2} - 3x - 2x + 6= 0}$

$\sf{ \implies \: (x - 3)(x - 2)= 0}$

$\sf{ \implies \: x= 3 \:,\: 2}$

x = 3 gives y = 4

x = 2 gives y = 5

Hence the required solutions are (3,4) & (2,5)

Since (3,4) is the vertex of the parabola

So another solution is (2,5)

FINAL ANSWER

Hence the correct option is

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

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