4. Write the equation of the line that passes through the point $A(1,3)$ and forms an isosceles triangle with the coordinate axes. Write all solutions.
4. We find that the segments cut off by the sought line on the coordinate axes are equal. We write $k_{1}=1 \mathrm{or} k_{2}=-1$. We also consider point $A$ in the equation of the sought line and write its equations as $y=x+2$ and $y=-x+4$.

$\mathrm{I}:$ method

Sketch 1 point

Observation $k_{1}=1$ or $k_{2}=-1 \ldots \ldots \ldots \ldots . . .1$ point

Writing the line $y_{1}=k_{1} x_{1}+n_{1} \ldots \ldots \ldots \ldots . . .1$ point

Calculation $n_{1}=2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$ point

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-10.jpg?height=534&width=708&top_left_y=224&top_left_x=1139)

Writing the line $y=x+2$. 1 point

Writing the second line $y=-x+4$

1 point

II: method

We calculate $n=4$ and write the equation $y=-x+4$. We also use the relationship $m=-n$ and calculate $n=2$ and write the equation $y=x+2$.

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-10.jpg?height=57&width=1636&top_left_y=1031&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-10.jpg?height=51&width=1639&top_left_y=1085&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-10.jpg?height=52&width=1642&top_left_y=1136&top_left_x=207)

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-10.jpg?height=46&width=1642&top_left_y=1188&top_left_x=204)

![](https://cdn.mathpix.com/cropped/2024_06_07_93a0a0489a8a740fd9e9g-10.jpg?height=60&width=1642&top_left_y=1232&top_left_x=204)