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Solve the system below using augmented matrix methods. Graph each solution set. Discuss the difference...

Solve the system below using augmented matrix methods. Graph each solution set. Discuss the difference between the graph of an equation in the system and the graph of the system's solution set.
4x1 - 3x2 = 8
5x1 - 4x2 = 8
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. The unique solution is x1 = __ and x2 = __. (Simplify your answers.)
B. There are infinitely many solutions. The solution is x1 = __ and x2 = t1 for any real number t. (Type an expression using t as the variable.)
C. There is no solution.

Answer

To solve the system using augmented matrix methods, we first write the augmented matrix:

Augmented Matrix:

[ 4 -3 | 8 ]
[ 5 -4 | 8 ]

Using row operations, we aim to get the matrix in row-echelon form:

  1. Multiply the first row by 5 and the second row by 4 to eliminate x1:

[ 20 -15 | 40 ]
[ 20 -16 | 32 ]

  1. Subtract the first row from the second row:

[ 20 -15 | 40 ]
[ 0 -1 | -8 ]

From the second row, we have -x2 = -8, so x2 = 8.

Substitute x2 = 8 into the first equation:

4x1 - 3(8) = 8
4x1 - 24 = 8
4x1 = 32
x1 = 8

Solution: The unique solution is x1 = 8 and x2 = 8.

Choice: A. The unique solution is x1 = 8 and x2 = 8. (Simplify your answers.)

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