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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
A. The unique solution is x₁ = □ and x₂ = □. (Simplify your answers.)
B. There are infinitely many solutions. The solution is x₁ = □ and x₂ = t, for any real number t. (Type an expression using t as the variable.)
C. There is no solution
Which graph represents the solution set for the given system of equations?

Answer

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

4 -3 | 8
5 -4 | 8

Perform row operations to reach 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
  1. Divide the second row by -1:

20 -15 | 40
0 1 | 8
  1. Substitute back to find x₁:

20x₁ - 15(8) = 40

20x₁ - 120 = 40

20x₁ = 160

x₁ = 8

Thus, the unique solution is x₁ = 8 and x₂ = 8.

Answer: A. The unique solution is x₁ = 8 and x₂ = 8.

Graph: The graph representing the solution set is a single point at (8, 8), which corresponds to option A.

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