Q.Find the equation of the line having x-intercept $2$ and y-intercept $-5$.

The $x$-intercept is found by setting $y=0$; here $3x-1=0$, giving $x=\tfrac13$.

The $x$-intercept is the point where the line crosses the $x$-axis, and every point on the $x$-axis has $y=0$. Substitute $y=0$ into the line:

$3x + 2(0) - 1 = 0 \;\Rightarrow\; 3x - 1 = 0 \;\Rightarrow\; x = \frac{1}{3}$

Rewriting $6x-2y+5=0$ in the form $y=mx+c$ gives $y=3x+\tfrac{5}{2}$, so the slope is $m=3$.

Start with the line:

$6x - 2y + 5 = 0.$

Solve for $y$:

$2y = 6x + 5 \;\Rightarrow\; y = 3x + \frac{5}{2}.$

Comparing with the slope-intercept form $y = mx + c$, the coefficient of $x$ is the slope:

$m = 3.$

(Equivalently, for a line $ax+by+c=0$ the slope is $-\dfrac{a}{b}=-\dfrac{6}{-2}=3$.)