Final answer:
The simultaneous equations 8x - 3y = 5 and -40x + 15y = -25 represent the same line, and thus have infinitely many solutions. Using the Gauss-Jordan method, we reach a simplified equation of x - (3/8)y = 5/8, from which any value for y can be chosen to solve for x.
Step-by-step explanation:
To solve the simultaneous equations 8x - 3y = 5 and -40x + 15y = -25 using the Gauss-Jordan method, we need to form an augmented matrix and then perform row operations to bring it to reduced row echelon form. We begin with the following matrix representation of the system:
- Row 1: [8 -3 | 5]
- Row 2: [-40 15 | -25]
We can initially see that the second equation is a multiple of the first one. Specifically, if we divide the second equation by -5, we obtain the first equation. This indicates that both equations actually represent the same line and therefore have infinitely many solutions. In a standard Gauss-Jordan elimination process, we would aim to have leading ones in each row and zeros in all other positions within the column of the leading one, but as both rows represent the same line, we can simplify to:
- Row 1: [1 -3/8 | 5/8]
- Row 2: [0 0 | 0]
From this, we can write the first equation as x - (3/8)y = 5/8, which is the simplified form of the line they represent. We can choose any value for y and then solve for x. Therefore, the solution is not a single point but a set of points lying on this line.