Find the equation of the line through the intersection of the lines 2x + 3y – 2 = 0 and x – 2y + 1 = 0 and having x-intercept equal to 3.
Suppose the given two lines intersect at a point P(x1, y1). Then, (x1, y1) satisfies each of the given equations.
2x + 3y – 2 = 0 …(i)
x – 2y + 1 = 0 …(ii)
Now, we find the point of intersection of eq. (i) and (ii)
Multiply the eq. (ii) by 2, we get
2x – 4y + 2 = 0 …(iii)
On subtracting eq. (iii) from (i), we get
2x + 3y – 2 – 2x + 4y – 2 = 0
⇒ 7y – 4 = 0
⇒ 7y = 4
Putting the value of y in eq. (ii), we get
Hence, the point of intersection P(x1, y1) is
Now, the equation of a line in intercept form is:
where a and b are the intercepts on the axis.
Given that: a = 3
⇒ bx + 3y = 3b …(i)
⇒ b + 12 = 21b
⇒ b – 21b = - 12
⇒ 20b = 12
Putting the value of ‘b’ in eq. (i), we get
⇒ 3x + 15y = 9
⇒ x + 5y = 3
Hence, the required equation of line is x + 5y = 3