Question

Using Cramer’s Rule, what is the value of x in the system of linear equations below? x+3y=16 3x+y=8

Answer 1

Using Cramer's Rule, the value of x in the given system of linear equations is 1.

Step-by-step explanation:

We can solve the given system of equations using Cramer's Rule. Cramer's Rule states that in a system of linear equations, if the determinant of the coefficient matrix is non-zero, then the system has a unique solution.

First, let's find the determinant of the coefficient matrix:

| 1 3 |

| 3 1 |

Det = (1*1) - (3*3) = -8

Since the determinant is non-zero (-8 ≠ 0), the system has a unique solution.

Next, let's find the determinant of the x-column matrix:

| 16 3 |

| 8 1 |

Det(x) = (16*1) - (3*8) = -8

Finally, we can find the value of x by dividing the determinant of the x-column matrix by the determinant of the coefficient matrix: x = Det(x) / Det = (-8) / (-8) = 1.

Answer 2

Cramer's rule works as follows:
x+3y=16
3x+y=8
Then
x=Dx/D
y=Dy/D
where Dx,Dy,D are 2x2 matrices formed from of coefficients and right hand side.
D=
1 3
3 1
=1-9=-8
Dx=matrix D with first column replaced by the vector [16,8]=
16,3
8 1
=16-24
=-8

Dy=matrix D with second column replaced by the vector [16,8]=
1 16
3 8
=8-48
=-40

Therefore
x=-8/-8=1
y=--40/-8=5

Check:
x+3y=1+3(5)=16
3x+y=3(1)+5=8 ok.