1800. In the "Fundamentals of Chemistry" by D. I. Mendeleev, data on the solubility of sodium nitrate $\mathrm{NaNO}_{3}$ depending on the temperature of water are provided. The following number of conditional parts of $\mathrm{NaNO}_{3}$ dissolve in 100 parts of water at the corresponding temperatures:

| Temperature ${ }^{\circ} \mathrm{C}$ | 0 | 4 | 10 | 15 | 21 | 29 | 36 | 51 | 68 |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| Amount of $\mathrm{NaNO}_{3}$ | 66.7 | 71.0 | 76.3 | 80.6 | 85.7 | 92.9 | 99.4 | 113.6 | 125.1 |

Assuming that the amount of $\mathrm{NaNO}_{3}(Y)$ that dissolves in 100 parts of water depends linearly on the temperature $(X)$ of the solution, find the parameters $a$ and $b$ in the formula $y=a x+b$ using the method of least squares.
Solution. To find the parameters $a$ and $b$ using the least squares method, it is necessary to solve system (3), in which in this case $x_{i}$ is the temperature of the solution; $y_{i}=$ the amount of sodium nitrate that dissolves in $100 \Omega$ of water at the given temperature; $\alpha_{1}=a ; \alpha_{2}=b$. The coefficients of this system are calculated according to the following scheme:

| $x_{i}$ | $y_{\imath}$ | $x_{i}^{2}$ | $x_{\imath} y$ |
| :---: | :---: | :---: | :---: |
| 0 | 66.7 | 0 | 0 |
| 4 | 71.0 | 16 | 284.0 |
| 10 | 76.3 | 100 | 763.0 |
| 15 | 80.6 | 225 | 1209.0 |
| 21 | 85.7 | 441 | 1799.7 |
| 29 | 92.9 | 841 | 2694.1 |
| 36 | 39.4 | 1296 | 3578.4 |
| 51 | 113.6 | 2601 | 5793.6 |
| 68 | 1251 | 4624 | 8506.8 |
| Sum 234 | 811.3 | 10144 | 24628.6 |

Thus, to determine $a$ and $b$, it is necessary to solve the following system of two equations with two unknowns:

$$
\left\{\begin{aligned}
10144 a+234 b & =24628.6 \\
234 a+9 b & =811.3
\end{aligned}\right.
$$

Multiply the left and right sides of the second equation by 26 and subtract from the first equation of the system. Then we get $4060 a=3534.8$. From this, $a=0.87 ; \quad 9 b=811.3-234 a=811.3-203.58=607.72 ; b=57.5$. Therefore, the dependence of $y$ on $x$ is $y=0.87 x+57.5$.