Question

2×+3y=13 4×-y=5 su cramen

Answer 1

To solve the system of equations using Cramer's Rule, we first express the system in matrix form. The given system of equations is:

1. 2x+3y=13
2. 4x−y=5

We can represent this system as a matrix equation AX=B, where:

`A = \left[\begin{array}{ccc}2&4\\3&-1\end{array}\right]`

`X = \left[\begin{array}{ccc}x\\y\end{array}\right]`

`B = \left[\begin{array}{ccc}13\\5\end{array}\right]`

Cramer's Rule states that if ∣A∣ ≠ 0, the solution to the system is given by
`x = (|A_(x)| )/(|A|)` and
`y = (|A_(y)| )/(|A|)` , where ∣A∣ is the determinant of matrix
`A , A_(x)` is the matrix obtained by replacing the first column of A with matrix B, and A is obtained by replacing the second column of A with B.

The determinant of matrix A is calculated as

∣A∣=(2×(−1))−(4×3)=−10.

Now, we substitute B for the respective columns to find
`|A_(x) |` and
`|A_(y) |`:

`|A_(x) | = \left[\begin{array}{ccc}13&3\\5&-1\\\end{array}\right]`

`|A_(y) | = \left[\begin{array}{ccc}2&13\\4&5\end{array}\right]`

Now, we can find the determinants
`|A_(x)| = (13 *(-1))- (5*3) = -40` and
`|A_(y)| = (2 *5)- (4*13) = -46`

Finally, we find the solutions:

`x = (|A_(x) |)/(A) = (-40)/(-10) = 4`

`y = (|A_(y) |)/(A) = (-46)/(-10) = 4.6`

Therefore, the solution to the system of equations is x=4 and y=4.6.