Problem Statement
Find the equation of the line passing through the point of intersection of the lines $2x + 3y - 4 = 0$ and $x - 5y + 7 = 0$ and having its $x$-intercept equal to $-4$.
Verified Solution & Marking Scheme
Write the Family of Lines Equation
Any line passing through the intersection of the two lines is of the form:
$(2x + 3y - 4) + \lambda (x - 5y + 7) = 0$
$(2 + \lambda)x + (3 - 5\lambda)y + (7\lambda - 4) = 0$ ...(1)
Apply the x-intercept Condition
The line has an $x$-intercept of $-4$, which means the point $(-4, 0)$ lies on the line.
Substituting $x = -4$ and $y = 0$ into (1):
$(2 + \lambda)(-4) + (3 - 5\lambda)(0) + (7\lambda - 4) = 0$
$-8 - 4\lambda + 7\lambda - 4 = 0$
$3\lambda - 12 = 0 \implies 3\lambda = 12 \implies \lambda = 4$
Substitute λ = 4 into Equation (1)
$(2 + 4)x + (3 - 5(4))y + (7(4) - 4) = 0$
$6x + (3 - 20)y + (28 - 4) = 0$
$6x - 17y + 24 = 0$.