Question

Two systems of equations are given below For each system, choose the best description of its solution If applicable, give the solution 7 System The system has no solution The system has a unique solution 5x-*= -1 5x+y=1 The system has infinitely many solutions Systeme The system has no solution The system has a unique solution: *+ 2y 13 -* + 2y = 7 The system has infinitely many solutions.

Answer 1

`5x\text{ - y = -1 and -5x + y = 1}`

If we sum both equations, we have the next result:

`0\text{ = 0}`

Since we have this, we can say that the system has infinite solutions. We sum both equations, and we finally get that 0 = 0. In this case, the system has infinite solutions.

All these solutions are expressed by (solving for y):

`y=\text{ 1 + 5x}`

For example, for a value of x = 1, y is a function of x; then, y = 1 + 5 = 6, or (1, 6), and so on.

For the next system of equations:

`\begin{gathered} x\text{ + 2y = 13} \\ -x\text{ + 2y = 7} \end{gathered}`

Adding both equations, we finally have:

`4y\text{ = 20}\Rightarrow\text{ y = 5}`

Then, solving for x, we have (using the first equation):

`x\text{ + 2(5) = 13 }\Rightarrow x\text{ = 13 - 10 }\Rightarrow x\text{ = 3}`

Then, this last system has a unique solution, which is (3, 5) or x = 3 and y = 5.