Answer :
The equation of a line that is parallel to 6x - 13y = 25 and passes through the point of intersection of 3x2y = 8 and 5x - 11y = 1 is 6x-13y = 19/15.
The equation of a line in the form of ax + by = c is given by
c = ax + by
Therefore, the equation of the line through the intersection of 3x2y=8, 5x-11y=1=0 and parallel to 6x-13y=25 is given by
c = 6x-13y
To solve for c, we substitute the intersection point of 3x2y=8 and 5x-11y=1=0 into the equation.
At the intersection point, x = 8/15 and y = -1/15
Substituting these values into the equation, we get
c = 6(8/15) - 13(-1/15)
c = 32/15 - 13/15
c = 19/15
Therefore, the equation of the line is 6x-13y = 19/15.
Alternatively, we can use the point-slope form of the equation of a line.
The point-slope form is given by
[tex]y - y1 = m(x - x1)[/tex]
where (x1, y1) is a point on the line and m is the slope of the line.
The slope of the line, m, can be found by substituting the values of x and y from the intersection point into the equation 6x-13y=25.
At the intersection point, x = 8/15 and y = -1/15
Substituting these values into the equation, we get
[tex]6(8/15) - 13(-1/15) = 25[/tex]
[tex]48/15 - 13/15 = 2535/15 = 2535 = 15*25m = 35/15[/tex]
Therefore, the equation of the line is given by
[tex]y - (-1/15) = (35/15)(x - 8/15)[/tex]
Simplifying, we get
[tex]y + 1/15 = 35/15x - 8/15[/tex]
[tex]15y + 1 = 35x - 815y = 35x - 9[/tex]
[tex]y = (35/15)x - (9/15)[/tex]
Therefore, the equation of the line is 6x-13y = 19/15.
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