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