Find the equation of the line passing through the point of intersection of the lines 4 x + 7 y – 3 = 0 and 2 x – 3 y + 1 = 0 that has equal intercepts on the axes.
The line is passing through the point of intersection of the lines,
4 x + 7 y − 3 = 0 (1)
2 x − 3 y + 1 = 0 (2)
Solve equation (1) and equation (2) for the point of intersection.
Substitute the value of y as 2 x + 1 3 from equation (2) to equation (1).
4 x + 7 ( 2 x + 1 3 ) − 3 = 0 4 x × 3 + 14 x + 7 − 3 × 3 3 = 0 12 x + 14 x + 7 − 9 = 0 26 x − 2 = 0
Further simplify the above expression.
x = 2 26 = 1 13
Substitute the value of x in equation (1).
4 1 13 + 7 y − 3 = 0 4 + 91 y − 39 13 = 0 91 y = 35 y = 35 91 = 5 13
The point of intersection of the two lines is ( 1 13 , 5 13 ) .
The formula for equation of line making an intercept of a and b on x axis and y axis respectively is given by,
x a + y b = 1
For equal intercepts,
a = b x a + y a = 1 x + y = a (3)
Substitute the values of x and y as 1 13 , 5 13 in equation
1 13 + 5 13 = a a = 6 13
Substitute the value of a in equation (3).
x + y = 6 13 13 x + 13 y = 6 13 x + 13 y − 6 = 0
Thus, the equation of line passing through the intersection of the lines 4 x + 7 y − 3 = 0 and 2 x − 3 y + 1 = 0 , making an equal intercept with the axes is 13 x + 13 y − 6 = 0 .
The line is passing through the point of intersection of the lines,
Solve equation (1) and equation (2) for the point of intersection.
Substitute the value of
Further simplify the above expression.
Substitute the value of
The point of intersection of the two lines is
The formula for equation of line making an intercept of
For equal intercepts,
Substitute the values of
Substitute the value of
Thus, the equation of line passing through the intersection of the lines
