Final answer:
The equation of the new line, we first determine the intercepts of the original line 2x + 3y + 11 = 0, then multiply these intercepts by 3 to get the intercepts of the new line. Using the intercept form of the equation, we derive the new line's equation as 2x + 6y + 33 = 0.
Step-by-step explanation:
To find the equation of a line whose intercepts on the axes are thrice as long as those of the given line
2x + 3y + 11 = 0,
we need to determine the x-intercept and the y-intercept of the given line.
Setting y to 0,
we find the x-intercept by solving 2x + 11 = 0,
which gives x = -11/2.
Setting x to 0,
we find the y-intercept by solving 3y + 11 = 0, which gives y = -11/3.
Therefore, the x-intercept is at (-11/2, 0) and the y-intercept is at (0, -11/3).
To find the intercepts of the new line that are thrice as long, we simply multiply these intercepts by 3.
The new x-intercept is (-11/2) * 3 = -33/2 and the new y-intercept is (-11/3) * 3 = -11.
Using the intercept form of the line equation
\(\frac{x}{a} + \frac{y}{b} = 1\),
where a is the x-intercept and b is the y-intercept, our new line equation becomes \(\frac{x}{-33/2} + \frac{y}{-11} = 1\).
By simplifying this, we have the equation of the new line as 2x + 6y + 33 = 0.