5x5 matrix with a calculator that only does 4x4

Use cofactors and solve it by Laplace expansion.
Here's a messy way.

Solve the 4 x 4 system you get by setting the 5th variable to 0 and only considering the first 4 equations.

Also solve the 4 x 4 system you get by setting the 5th variable to 1 and all the constants to 0 and only considering the first 4 equations.

If the first has solution (a1,a2,a3,a4) and the second (b1,b2,b3,b4), then solutions to the first 4 equations will be of the form (a1+t\*b1,a2+t\*b2,a3+t\*b3,a4+t\*b4,t).

Use the 5th equation to solve for t.

Edit: it's possible the first step won't have a solution, if so try a different variable in the role of the 5th variable.
You could solve the first column into 1 0 0 0 0 then put row 2-5 column 2-5+c into your calculator
Row reduce by hand to get one row/column to be all zeroes except on the diagonal. Then use the calculator to do the rest.
Is there a reason why you cannot use better software? This specific calculator is required for a test, or something like that?

We see these questions regularly on this subreddit, and the correct answer is always to use the correct tool to finish the job. When someone asks *how do I use a hammer to turn a screw* the answer is *borrow a screwdriver*.