A telecommunications technician is mapping a new fiber-optic cable that must be laid parallel to an existing line defined by the equation $y = 3x + 1$. The new cable must pass through a connection point located at (4, 2). To begin determining the equation for the new cable, which value from the given information must be identified as the slope ($m$)?

A land surveyor is mapping a new boundary line that must be parallel to an existing fence line represented by the equation $y = 3x + 1$. The new line must pass precisely through a survey marker located at ({}4, 2). Arrange the steps the surveyor must follow to determine the equation of the new line in slope-intercept form.

An interior designer is placing parallel decorative strips on a wall. The first strip follows the line $y = 3x + 1$, and the second strip must pass through the point (4, 2). To find the equation for the second strip, the designer should substitute 4 for the $y_1$ value and 2 for the $x_1$ value in the point-slope formula $y - y_1 = m(x - x_1)$.

A civil engineer is designing a new drainage line that must run parallel to an existing pipe described by the equation $y = 3x + 1$. The new line is required to pass through a specific access point located at ({}4, 2). To construct the equation for the new line, the engineer uses the point-slope formula: $y - y_1 = m(x - x_1)$. Match the following variables and formulas with the correct numerical values or algebraic representations they correspond to in this engineering scenario.

Surveying Parallel Boundaries

Determining Parallel Slope for Solar Panel Rows

A network infrastructure planner is designing a new fiber-optic cable route that must run parallel to an existing main line represented by the equation $y = 3x + 1$. The new cable route must pass directly through a local distribution hub located at the coordinates ({}4, 2). To map the new route in the central system, the planner must enter its path in slope-intercept form, $y = mx + b$. Fill in the blank to complete the expression representing the right side of the equation for this new route: