(1, 6) is a solution to −x+y=5 and y=(x−3)^2+2

Question

Responses:

True

False

GPT-4o mini · Answer

To determine whether (1, 6) is a solution to the equations $ -x + y = 5 $ and $ y = (x - 3)^2 + 2 $, we can substitute $ x = 1 $ and $ y = 6 $ into both equations.

For the first equation $ -x + y = 5 $: \[ -1 + 6 = 5 \ 5 = 5 \] This equation is true.
For the second equation $ y = (x - 3)^2 + 2 $: \[ 6 = (1 - 3)^2 + 2 \ 6 = (-2)^2 + 2 \ 6 = 4 + 2 \ 6 = 6 \] This equation is also true.

Since (1, 6) satisfies both equations, the statement is:

True

Yes. · Answer

A system of equations with one linear function and one quadratic function could have infinitely many solutions. Responses True False

GPT-4o mini · Answer

A system of equations with one linear function and one quadratic function could have one of the following scenarios:

No solutions: This occurs when the line and the parabola do not intersect.
One solution: This occurs when the line is tangent to the parabola.
Two solutions: This occurs when the line intersects the parabola at two points.

However, a linear function and a quadratic function cannot produce infinitely many solutions together. Infinitely many solutions would imply the two functions are the same line at all points, which is not possible with one being linear and the other being quadratic.

Thus, the statement is:

False

Yes. · Answer

(0, 3) is a solution to 1/3x+y=3 and 2y =−2/3x+10 Responses True False

GPT-4o mini · Answer

To determine whether (0, 3) is a solution to the equations $ \frac{1}{3}x + y = 3 $ and $ 2y = -\frac{2}{3}x + 10 $, we will substitute $ x = 0 $ and $ y = 3 $ into both equations.

For the first equation $ \frac{1}{3}x + y = 3 $: \[ \frac{1}{3}(0) + 3 = 3 \ 0 + 3 = 3 \ 3 = 3 \] This equation is true.
For the second equation $ 2y = -\frac{2}{3}x + 10 $: \[ 2(3) = -\frac{2}{3}(0) + 10 \ 6 = 0 + 10 \ 6 = 10 \] This equation is false.

Since (0, 3) does not satisfy the second equation, the overall result is:

False

Yes. · Answer

A system of equations with two linear functions could have infinitely many solutions.(1 point) Responses True False

GPT-4o mini · Answer

A system of equations with two linear functions can result in infinitely many solutions if the two equations represent the same line. This occurs when both equations are equivalent, meaning they have the same slope and y-intercept.

Thus, the statement is:

True

Yes. · Answer

Use the following scenario to solve questions 5 - 7.

Santiago is working at his mom's daycare this summer. His mom asked him to check out the library book sale to look for books they might be able to use in the daycare. When he gets to the book sale, he sees that board books are $1.50 and soft cover books are $0.50. He picks out a variety of books and spends $20. Which equation represents the situation described?

Responses

1.50x + 0.50x = 20

2x = 20

1.50x + 0.50y = 20

20 - 2x = 0

GPT-4o mini · Answer

In this scenario, let $ x $ represent the number of board books and $ y $ represent the number of softcover books. The costs of the respective books are $ 1.50 $ dollars for each board book and $ 0.50 $ dollars for each softcover book. Santiago spends a total of $ 20 $ dollars. The equation that represents the total amount spent on both types of books is: $$ 1.50x + 0.50y = 20 $$ Thus, the correct answer is: **1.50x + 0.50y = 20**

Yes. · Answer

At the book sale, board books are $1.50 and soft cover books are $0.50. True or false: Santiago bought 10 board books and 5 soft cover books for exactly $20. Responses True False

GPT-4o mini · Answer

To determine if Santiago's purchase of 10 board books and 5 softcover books amounts to exactly $20, we can calculate the total cost.

Cost of board books:
- 10 board books at $1.50 each:
  \[ 10 \times 1.50 = 15.00 \]
Cost of softcover books:
- 5 softcover books at $0.50 each:
  \[ 5 \times 0.50 = 2.50 \]
Total cost: \[ 15.00 + 2.50 = 17.50 \]

Since the total cost of $17.50 is not equal to $20, the statement is:

False.