A landscape architect is designing a new stone walkway that must be perpendicular to a straight garden edge modeled by the equation $y = 3x + 1$. The walkway must pass through a specific fountain located at the coordinates (4, 2). Arrange the following steps of the 'five-step algebraic mapping method' in the correct order to determine the final equation for the walkway's path.

A civil engineer is mapping out a new drainage pipe that must be laid exactly perpendicular to an existing underground utility line. The path of the existing utility line is defined on the site plan by the equation $y = 3x + 1$. Based on the algebraic rules for perpendicular paths, what specific slope must the engineer recall and use for the new drainage pipe?

A telecommunications technician is mapping a new fiber optic branch that must run perpendicular to a primary line represented by the equation $y = 3x + 1$. The branch must connect to a hub located at the coordinates (4, 2). Based on the five-step algebraic mapping method, match each component of the layout plan with its corresponding mathematical value.

In a construction layout, if a primary reference line is defined by the equation $y = 3x + 1$, the slope of a second line intended to run exactly perpendicular to it must be ${}-\frac{1}{3}$.

Precision Infrastructure: Perpendicular Utility Path

A geographic information system (GIS) specialist is mapping a new secondary power line that must run exactly perpendicular to an existing high-voltage transmission corridor modeled by the equation $y = 3x + 1$. The secondary line must connect to a grid substation located at coordinates (4, 2). Using the five-step algebraic mapping method, the specialist determines that the final equation of the secondary line in slope-intercept form is $y = -\frac{1}{3}x + b$. What is the value of the y-interc

Procedural Verification: Mapping Perpendicular Utility Paths