1. The points $A(-2,1)$ and $B(1,-11)$ are in a Cartesian coordinate system.

(a) Determine the coordinates of point $C$, which lies on the x-axis, such that the area of triangle $ABC$ is 64.5.

(b) The graph of a linear function passes through point $A$. If the slope were 3 units greater, the graph would pass through point $B$. Determine the equation of this linear function.
1. 

(a) We are looking for the abscissa of point $C(x, 0)$. We use the formula for calculating the area

$$
S=\frac{1}{2} \cdot o \cdot\left|\begin{array}{ll}
x_{2}-x_{1} & y_{2}-y_{1} \\
x_{3}-x_{1} & y_{3}-y_{1}
\end{array}\right|
$$

we substitute the coordinates of points $A, B, C$ and multiply by 2 to get $129=o \cdot\left|\begin{array}{cc}3 & -12 \\ x+2 & -1\end{array}\right|$. We calculate the determinant and get the relationship $129=o \cdot(12 x+21)$. If the orientation $o=+1$, we get the solution $x_{1}=9$, and if the orientation $o=-1$, we get the solution $x_{2}=-\frac{25}{2}$. For point $C$, we thus get two possibilities $C_{1}=(9,0)$ and $C_{2}=\left(-\frac{25}{2}, 0\right)$.

(b) We are looking for the linear function $f(x)=k \cdot x+n$. Considering that the graph passes through point $A$, we get the equation $1=-2 k+n$. If $k$ is 3 greater, the graph passes through $B$ and we get the equation $-14=$ $k+n$. We solve the obtained system of two linear equations with two unknowns and get $f(x)=$ $-5 x-9$.

Writing or using the formula for calculating the area $S=\frac{1}{2} \cdot o \cdot\left|\begin{array}{ll}x_{2}-x_{1} & y_{2}-y_{1} \\ x_{3}-x_{1} & y_{3}-y_{1}\end{array}\right| \ldots \ldots .1$ point

Reformulating into the equation $129=o \cdot(12 x+21)$ or $129=|12 x+21| \ldots \ldots \ldots \ldots \ldots \ldots$.

![](https://cdn.mathpix.com/cropped/2024_06_07_3b81dd7f2d1678befe89g-12.jpg?height=60&width=1727&top_left_y=2443&top_left_x=176)

![](https://cdn.mathpix.com/cropped/2024_06_07_3b81dd7f2d1678befe89g-12.jpg?height=60&width=1727&top_left_y=2494&top_left_x=176)

Writing or using the equation for the linear function $f(x)=k \cdot x+n \ldots \ldots \ldots \ldots \ldots \ldots . . .$.

![](https://cdn.mathpix.com/cropped/2024_06_07_3b81dd7f2d1678befe89g-12.jpg?height=49&width=1727&top_left_y=2648&top_left_x=176)

Equation if $k$ is 3 greater and the graph passes through $B:-14=k+n \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots . \ldots$.
Solving the system of equations ..... 1 point
Writing $f(x)=-5 x-9$ ..... 1 point