To find point
A, we need to solve the equations:
‌3x+y−4=0‌x−y=0The second equation gives us
y=x. Substitute
y=x into the first equation:
‌3x+x−4=0‌4x−4=0‌4x=4‌x=1Substituting
x=1 into
y=x, we get
y=1.
Thus, the coordinates of
A are
(1,1).
Next, consider the line
x−3y+5=0 which has a slope of
‌.
Let
m be the slope of the required line. Since the line makes an angle of
45∘ with the given line and has a negative slope, we use the angle formula:
tan‌45∘=|‌|Since
tan‌45∘=1, we have:
1=|‌|This simplifies to:
|3+m|=|1−3m|Solving
3+m=±(1−3m) gives two possibilities:
‌3+m=1−3m‌4m=−2‌‌⇒‌‌m=−‌‌3+m=−1+3m‌−2m=−4‌‌⇒‌‌m=2Since the slope must be negative, we choose
m=−‌.
The equation of the line using point-slope form is:
y−1=−‌(x−1)Simplifying:
‌2(y−1)=−(x−1)‌2y−2=−x+1‌x+2y=3Thus, the required equation of the line is:
x+2y=3