1. The line passes through the intersection of the lines $11 x+3 y-7=0, 12 x+y-19=0$ and the midpoint of the segment with endpoints $A(3,-2)$ and $B(-1,6)$. Write the equation of this line $\mathrm{v}$ in all three forms and name them. Draw the line.

(10 points)

![](https://cdn.mathpix.com/cropped/2024_06_07_115a7be78276e1d4adcag-04.jpg?height=856&width=868&top_left_y=857&top_left_x=640)
1. After solving the system of equations using any method, we get the intersection point $P(2,-5)$. We determine the midpoint of the segment $A B: S(1,2)$. We calculate the slope $k=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{2+5}{1-2}=-7$.

We substitute the data into the equation of the line, for example: $y-y_{1}=k\left(x-x_{1}\right)$ and after rearranging, we get $y=-7 x+9$. We also convert it to the other two forms. We draw the line.

![](https://cdn.mathpix.com/cropped/2024_06_07_115a7be78276e1d4adcag-13.jpg?height=845&width=851&top_left_y=551&top_left_x=614)
Solving the system of equations ..... $1^{*}$ point
Writing the intersection of the lines $P(2,-5)$ ..... $1+1$ points
Writing the midpoint of the segment $A B S(1,2)$ ..... 1 point
Calculating the slope $k=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=\frac{2+5}{1-2}=-7$ ..... 1 point
Substituting the data into the equation of the line ..... 1 point
Writing the explicit form of the equation of the line $y=-7 x+9$ ..... 1 point
Writing the implicit form of the equation of the line $7 x+y-9=0$ ..... 1 point
Writing the intercept form of the equation of the line $\frac{x}{\frac{9}{7}}+\frac{y}{9}=1$ ..... 1 point
Drawing the line ..... 1 point