B1. For the linear function $f$, the following holds:

a) If its slope is increased by 1 and the y-intercept by 2, its value at some $x_{0}$ increases by 5.

b) If this $x_{0}$ is decreased by 2, the function $f$ has six times the value at $x_{0}$.

c) The graph of the function $f$ passes through the point $A(4,-3)$.

Write the appropriate relationships and calculate the formula for $f(x)$ and the value of $x_{0}$.
B1. A linear function has the formula $f(x)=k \cdot x+n$.

At $x_{0}$ it has the value $y_{0}=f\left(x_{0}\right)=k \cdot x_{0}+n$. If the slope $k$ is increased by 1 and the y-intercept $n$ by 2, we get the function $g(x)=(k+1) \cdot x+n+2$. Substituting $x_{0}$, we get $g\left(x_{0}\right)=(k+1) \cdot x_{0}+n+2=$ $k \cdot x_{0}+x_{0}+n+2=y_{0}+x_{0}+2$. Since in this case the value of the function $y_{0}=f\left(x_{0}\right)$ increases by 5, we get $x_{0}+2=5$ and $x_{0}=3$.

At $x_{0}-2=3-2=1$, the function $f(x)$ has the value $f(1)=k \cdot 1+n=k+n$. Considering that this is six times the value of $y_{0}=f\left(x_{0}\right)=f(3)=k \cdot 3+n$, we get the equation $k+n=6(3 k+n)$. This simplifies to $17 k+5 n=0$.

If the graph of the function $f(x)$ passes through the point $A(4,-3)$, then $f(4)=-3$. We get the equation $4 k+n=-3$. Solving the system of equations $17 k+5 n=0$ and $4 k+n=-3$, we get $k=-5$ and $n=17$. The sought linear function thus has the formula $f(x)=-5 x+17$.

Considering the condition in a) for example, $g(x)=(k+1) \cdot x+n+2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-20.jpg?height=57&width=1727&top_left_y=1896&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-20.jpg?height=54&width=1727&top_left_y=1943&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-20.jpg?height=57&width=1727&top_left_y=1996&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-20.jpg?height=57&width=1727&top_left_y=2045&top_left_x=176)

Writing the appropriate equation for condition c) for example, $4 k+n=-3$...................................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-20.jpg?height=48&width=1727&top_left_y=2146&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_37ea02f5a530d200baf4g-20.jpg?height=60&width=1731&top_left_y=2192&top_left_x=174)