Question

What is the maximum number of possible solutions for the system shown below?


`x^2+4y^2=64\\x+y=5`

Answer 1

The maximum number of solutions that the system can have are 2

Explanation:

We have a system of equations composed of an ellipse of equation
`x ^ 2 + 4y ^ 2 = 64` and a line of equation
`x + y = 5.`

The solution to the system of equations gives information about when the values of both equations coincide. In other words, the solutions of the system of equations give information about the number of times that the line
`x + y = 5`. touch or intercept the graph of the ellipse
`x ^ 2 + 4y ^ 2 = 64`

Graphically you can verify that a line can only touch an ellipse once or can intercept an ellipse a maximum of two times, or never touch it

Therefore, the maximum number of solutions that the system can have are 2.

Answer 2

2

Explanation:

The first equation is that of a an ellipse. The second equation is that of a line.

Attached is the graphs of both of these equations.

If you think about it, there can only be 2 possible ways of solutions (intersection points) of an ellipse and a line.

1. The line will not intersect the ellipse at all, so no solution

2. The line will intersect the ellipse at 2 points maximum

So, we can clearly see from the reasoning that the maximum number of possible solutions would be 2. The graph attached confirms this as well.