(a) Given that the lines 7 x+5 y=4, x+k y=3, and 5 x+y+k=0...

(a) Given that the lines $7 x+5 y=4, x+k y=3,$ and $5 x+y+k=0$ are concurrent (pass through a common point), what are the possible values for $k ?$ (b) Check that your answers are reasonable: For each value of $k$ that you find, use a graphing utility to draw the three lines. Do they appear to be concurrent?

(a) Given that the lines $7 x+5 y=4, x+k y=3,$ and $5 x+y+k=0$ are concurrent (pass through a common point), what are the possible values for $k ?$
(b) Check that your answers are reasonable: For each value of $k$ that you find, use a graphing utility to draw the three lines. Do they appear to be concurrent?

Key Concepts

Graphical Verification

Graphical verification involves using a graphing utility or tool to plot the lines and visually confirm that they intersect at a single point. This method serves as an additional check to ensure that the mathematically determined conditions for concurrency are reasonable and correctly applied.

Parameter Determination

When a linear equation contains a parameter, determining the value of that parameter so that the lines meet at a common point involves imposing the concurrency condition. This requires combining the equations in a way that yields a condition for the parameter, ensuring that the resulting system of equations has a unique solution at the point of intersection.

Concurrency of Lines

Concurrency refers to the property of three or more lines meeting at a single point in a plane. When lines are concurrent, there exists a unique point that satisfies all their equations simultaneously. This concept is fundamental in geometry and helps in understanding the intersection properties of linear equations.

Systems of Linear Equations

A system of linear equations involves multiple equations that contain the same set of variables. Solving such a system requires finding a common solution that satisfies all equations. In the context of lines, the solution represents the point of intersection. Techniques such as substitution or elimination can be employed to solve these systems.

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