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Parametric equations provide a way to define a line in three-dimensional space using a single parameter. For the line \( \ell_1 : (x, y, z) = (2,3,-1) + n(1,1,2) \), the vector \((2,3,-1)\) represents a point on the line, and \((1,1,2)\) indicates the direction the line is heading. The parameter \( n \) is a variable that, when varied, traces out the line. By adjusting \( n \), you can find any other point on the line.
Similarly, the line \( \ell_2 : (x, y, z) = (7,7,2) + r(-2,-1,3) \) uses the parameter \( r \) to traverse along its direction vector \((-2,-1,3)\). Parametric equations are particularly useful for describing geometric shapes because they give you flexibility and precision. You can intuitively imagine moving along the line by adjusting the parameter instead of calculating every point.

When two lines intersect, setting their equations equal generates a system of equations. This system forms because you are essentially looking for a common set of points that satisfy both lines' equations. For the problem at hand, you would set the equations of lines \( \ell_1 \) and \( \ell_2 \) equal:

• \( 2 + n = 7 - 2r \)
• \( 3 + n = 7 - r \)
• \( -1 + 2n = 2 + 3r \)

Solving this system allows you to find the values of \( n \) and \( r \) where the lines intersect. A system of equations provides a structured method to discover points, such as intersection points, where different geometries meet. You solve it by finding a common solution to all equations in the system, representing points occupying the same space on both lines.

To solve the system of equations derived from setting parametric equations equal, familiar algebraic techniques come into play. The goal is to isolate each variable and find consistent values for each parameter, ensuring the solution satisfies all equations. Begin by using algebraic manipulation on the first two equations:

• Solve \( 2 + n = 7 - 2r \) to find an expression for \( n \): \( n = 5 - 2r \).
• Solve \( 3 + n = 7 - r \) to express \( n \) another way: \( n = 4 - r \).
• Set the expressions for \( n \) equal: \( 5 - 2r = 4 - r \) and solve for \( r \).

Once \( r \) is found (\( r = 1 \)), substitute it back to find \( n \) (\( n = 3 \)). Finally, verify the solution satisfies the third equation, which confirms correctness and consistency. Solving algebraic equations like this requires careful manipulation and checking each substitution step to ensure accuracy across the entire system of equations.