73. Find the equation of the line that forms an angle of $135^{\circ}$ with the $O x$ axis and intersects the $O y$ axis at the point $(0 ; 5)$. Determine whether this line passes through the points $A(2 ; 3)$ and $B(2 ;-3)$. Plot the line.
Solution. From the condition of the problem, it follows that the segment cut off by the line on the ordinate axis, $b=5$, the slope $k=\operatorname{tg} 135^{\circ}=-1$. Therefore, by formula (1) we have

$$
y=-x+5
$$

Substituting the coordinates of point $A$ into the desired equation of the line instead of the current coordinates, we get

$$
3=-2+5, \text { i.e., } 3=3
$$

Since the coordinates of point $A$ satisfy the equation of the line, the line passes through this point.

Substituting the coordinates of point $B$ into the equation, we get

$$
-3 \neq-2+5, \text { i.e., }-3 \neq 3
$$

The coordinates of point $B$ do not satisfy the equation, therefore, the line does not pass through point $B$.

It is known that the position of a line is determined by two points lying on it. Therefore, to construct a line from its equation, we can find any two points whose coordinates satisfy this equation, plot them, and draw the line through them. The constructed line will be the desired one.

Let $x=1$, then $y=-1+5=4$, if $x=-2$, then $y=$ $=2+5=7$.

Thus, we have obtained two points $P(1 ; 4)$ and $Q(-2 ; 7)$. Let's plot these points and draw the line through them (Fig. 12).

Note that the equation of the given line contains a free term, therefore, this line intersects the coordinate axes. Let's find the points of intersection of the line with the coordinate axes and draw the desired line through them. This method of constructing a line is simpler.

To find the point of intersection of the line with the $O y$ axis, we need to set $x=0$ (any point lying on the ordinate axis has an abscissa equal to 0), then $y=5$. Let's denote this point by $C(0 ; 5)$. Now let $y=0$ (any point lying on the abscissa axis has an ordinate equal to 0), then $0=$ $=-x+5, x=5$. Let's denote this point by $D(5 ; 0)$. Let's plot these points and draw the desired line through them (see Fig. 12 ).

![](https://cdn.mathpix.com/cropped/2024_05_22_1b9a1c1720bae4af5bc5g-025.jpg?height=408&width=490&top_left_y=350&top_left_x=632)

Fig. 12