1. **Stating the problem:** Given points Tink M $(2, -7)$ and Oleh $(3, 4)$, and a vector equation $8x + 3y = 9$, find the projection of the vector from Tink M to Oleh onto the vector $(7, -5)$. 2. **Understanding the problem:** We want to find the projection of vector $\vec{A} = \overrightarrow{TinkM \, to \, Oleh}$ onto vector $\vec{B} = (7, -5)$. 3. **Calculate vector $\vec{A}$:** $$\vec{A} = (3 - 2, 4 - (-7)) = (1, 11)$$ 4. **Formula for projection:** The projection of $\vec{A}$ onto $\vec{B}$ is given by: $$\text{proj}_{\vec{B}} \vec{A} = \frac{\vec{A} \cdot \vec{B}}{\|\vec{B}\|^2} \vec{B}$$ where $\vec{A} \cdot \vec{B}$ is the dot product and $\|\vec{B}\|$ is the magnitude of $\vec{B}$. 5. **Calculate dot product:** $$\vec{A} \cdot \vec{B} = (1)(7) + (11)(-5) = 7 - 55 = -48$$ 6. **Calculate magnitude squared of $\vec{B}$:** $$\|\vec{B}\|^2 = 7^2 + (-5)^2 = 49 + 25 = 74$$ 7. **Calculate projection vector:** $$\text{proj}_{\vec{B}} \vec{A} = \frac{-48}{74} (7, -5) = \left(\frac{-48 \times 7}{74}, \frac{-48 \times (-5)}{74}\right) = \left(\frac{-336}{74}, \frac{240}{74}\right)$$ 8. **Simplify fractions:** $$\frac{-336}{74} = \frac{-168}{37}, \quad \frac{240}{74} = \frac{120}{37}$$ 9. **Final answer:** The projection vector is: $$\boxed{\left(-\frac{168}{37}, \frac{120}{37}\right)}$$ This vector represents the shadow (projection) of $\vec{A}$ on $\vec{B}$.