2. Three points $A, B$, and $C$ are placed in a coordinate system in the plane. Their abscissas are the numbers $2, -3, 0$, and their ordinates are the numbers $-3, 2, 5$, not necessarily in that order. Determine the coordinates of these points if it is known:

- No point is in the III. quadrant.
- Point $A$ does not lie on a coordinate axis.
- The ordinate of point $B$ is equal to the sum of the ordinates of the other two points.
- The abscissa of point $C$ is equal to the ordinate of point $B$.
First method:

The student can solve the problem using the table method.

![](https://cdn.mathpix.com/cropped/2024_05_30_cbd21e468efb4566cfb2g-1.jpg?height=503&width=1602&top_left_y=2184&top_left_x=250)

No point is in the third quadrant, meaning that there is no possible combination of a negative abscissa and a negative ordinate.

![](https://cdn.mathpix.com/cropped/2024_05_30_cbd21e468efb4566cfb2g-2.jpg?height=509&width=1608&top_left_y=328&top_left_x=247)

Point $A$ does not lie on the coordinate axis, meaning that point $A$ cannot have an abscissa of 0.

|  |  | abscissa |  |  | ordinate |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  |  | 2 | -3 | 0 | 2 | -3 | 5 |
|  | A |  |  | - |  |  |  |
|  | B |  |  |  |  |  |  |
|  | C |  |  |  |  |  |  |
| ![](https://cdn.mathpix.com/cropped/2024_05_30_cbd21e468efb4566cfb2g-2.jpg?height=170&width=78&top_left_y=1202&top_left_x=337) | 2 |  |  |  |  |  |  |
|  | -3 |  | - |  |  |  |  |
|  | 5 |  |  |  |  |  |  |

The ordinate of point $B$ is equal to the sum of the ordinates of the remaining two points. The only possible ordinate for point $B$ is 2 because $-3 + 5 = 2$.

![](https://cdn.mathpix.com/cropped/2024_05_30_cbd21e468efb4566cfb2g-2.jpg?height=506&width=1605&top_left_y=1523&top_left_x=248)

The abscissa of point $C$ is equal to the ordinate of point $B$. This means that the abscissa of point $C$ is 2.

|  |  | abscissa |  |  | ordinate |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  |  | 2 | -3 | 0 | 2 | -3 | 5 |
|  | A | - |  | - | $\overline{-}$ |  |  |
|  | B | - |  |  | + | - | - |
|  | C | + | - | - | - |  |  |
| $\stackrel{\pi}{\pi}$  $\underset{\pi}{3}$  0 | 2 |  |  |  |  |  |  |
|  | -3 |  | - |  |  |  |  |
|  | 5 |  |  |  |  |  |  |

From the table, it is easy to see that the abscissa of point $B$ is 0, and the abscissa of point $A$ is -3.

|  |  | abscissa |  |  | ordinate |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  |  | 2 | -3 | 0 | 2 | -3 | 5 |
|  | A | - | + | - | $=$ |  |  |
|  | B | - | $\overline{-}$ | + | + | $\overline{-}$ | - |
|  | C | + | - | - | - |  |  |
| ![](https://cdn.mathpix.com/cropped/2024_05_30_cbd21e468efb4566cfb2g-3.jpg?height=174&width=78&top_left_y=617&top_left_x=339) | 2 |  |  |  |  |  |  |
|  | -3 |  | - |  |  |  |  |
|  | 5 |  |  |  |  |  |  |

Therefore, the coordinates of point $B$ are $B(0,2)$.

![](https://cdn.mathpix.com/cropped/2024_05_30_cbd21e468efb4566cfb2g-3.jpg?height=491&width=1602&top_left_y=925&top_left_x=247)

The coordinates of the points are $A(-3,5)$ and $C(2,-3)$.

|  |  | abscissa |  |  | ordinate |  |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  |  | 2 | -3 | 0 | 2 | -3 | 5 |
|  | A | - | + | - | - | - | + |
|  | B | - | - | + | + | - | - |
|  | C | + | - | - | - | + | - |
| ![](https://cdn.mathpix.com/cropped/2024_05_30_cbd21e468efb4566cfb2g-3.jpg?height=178&width=78&top_left_y=1831&top_left_x=339) | 2 | - | $\overline{-}$ | + |  |  |  |
|  | -3 | + | - | - |  |  |  |
|  | 5 | - | + | - |  |  |  |

- For each correctly determined coordinate of each point, 1 POINT should be awarded

TOTAL 6 POINTS

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