Systems of Equations with Parameters

Question

Systems of Equations with Parameters: For what value of k does the system of equations x+y=kx + y = kx+y=k and 2x−3y=62x - 3y = 62x−3y=6 have no solutions?

Answer 1

Solve the first equation for y:

y = - x + k (the slope of this line will always be -1, regardless of the value given to k.

Solve the 2nd equation for y:

3y = 2x - 6

y = 2x/3 - 2 (the slope of this line is 2/3.

For two linear equations to have no point of intersection (no solution to the system), the slopes of the two lines must be equal, i.e. the lines must be parallel.

Since the two given equations will never have the same slope, the lines will never be parallel, so the system will always have a solution.

Visit this graph, use the slider on k to see that the two equations always have a point of intersection.

desmos.com/calculator/hpazsk6el7

To visit the graph select the link, right-click, and choose "go to..." from the drop-down menu.