Question

Use the echelon method to solve the system of two equations in two unknowns. Check your answers. 5x – 3y = 1 8x - y = 13 Select the correct choice below and fill in any answer boxes within your choice. O A. The solution of the system is . (Simplify your answer. Type an ordered pair) OB. There are infinitely many solutions. The solution is ( O C. There is no solution y), where y is any real number

Answer 1

x = 2, y = 3

Explanation:

Given equations,

5x - 3y = 1,

8x - y = 13

Augmented matrix,

`A_b=\begin{bmatrix}5 & -3 |& 1 \\ 8 & -1 |& 13\end{bmatrix}`

`R_2\rightarrow (1)/(8)R_2`

`\begin{bmatrix}5 & -3 |& 1 \\ 1 & -1/8 |& 13/8\end{bmatrix}`

`R_1\rightarrow R_1-5R_2`

`\begin{bmatrix}0 & -19/8 |& -57/8 \\ 1 & -1/8 |& 13/8\end{bmatrix}`

`R_1\rightarrow (R_1)/(-19/8)`

`\begin{bmatrix}0 & 1 |& 3 \\ 1 & -1/8 |& 13/8\end{bmatrix}`

`R_2\rightarrow R_2+1/8R_1`

`\begin{bmatrix}0 & 1 |& 3 \\ 1 & 0 |& 2\end{bmatrix}`

`R_1` ↔
`R_2`

`\begin{bmatrix}1 & 0 |& 2 \\ 0 & 1 |& 3\end{bmatrix}`

`\implies x+0y = 2\implies x = 2`

Also,

`0x+y=3\implies y = 3`

Hence, the solution of the given system would be,

x = 2, y = 3