5. When solving the equation $\frac{10 x+6}{5}=3-1.6 x$, the student instead of the coefficient $1.6$ with the unknown $x$ on the right side of the equation wrote some other number. In this way, he obtained a value for the unknown $x$ that was 0.1 less than its actual value. What number did the student write instead of the coefficient 1.6?

## Problems worth 10 points:
5. When solving the equation $\frac{10 x+6}{5}=3-1.6 x$, a student wrote a different number instead of the coefficient 1.6 for the unknown $x$ on the right side of the equation. As a result, the student obtained a value for $x$ that was 0.1 less than its actual value. What number did the student write instead of the coefficient 1.6?

## Solution.

First, let's solve the given equation as it is written to find the actual solution:

$\frac{10 x+6}{5}=3-1.6 x \quad / \cdot 5$

$$
\begin{aligned}
& 10 x+6=15-8 x \\
& 1 \text { POINT } \\
& 10 x+8 x=15-6 \\
& 18 x=9 \quad /: 18 \\
& x=0.5 \quad 1 \text { POINT } \\
& \text { The student obtained a value that is } 0.1 \text { less, so } x=0.4 \text {. } 1 \text { POINT } \\
& \text { If we substitute this solution for } x \text {, and replace the number } 1.6 \text { with the unknown } y \text {, we get: } \\
& \frac{10 \cdot 0.4+6}{5}=3-y \cdot 0.4 \\
& \frac{10}{5}=3-0.4 y \\
& 2=3-0.4 y \quad \\
& 1 \text{POINT} \\
& 0.4 y=3-2 \\
& 0.4 y=1 \quad / \cdot 10 \quad 1-20 \\
& 4 y=10 \quad /: 4 \\
& y=2.5
\end{aligned}
$$

The student wrote the number 2.5 instead of 1.6.