Question

10) find x using Cramer's rule:x - 2y + 3z = 72x + y + z = 43x + 2y - 2z = - 10

Answer 1

Hello!

First, let's rewrite the system as a matrix:

$\begin{bmatrix}{1} & {-2} & {3} & {7} \\ {2} & {1} & {1} & {4} \\ {-3} & {2} & {-2} & {-10} \\ x & y & z & result{}\end{bmatrix}$

Let me explain:

• 1st, column: coefficients from x

,

• 2nd, column: coefficients from y

,

• 3rd ,column: coefficients from z

,

• 4th, column: results

Now, we must calculate the determinant of the matrix 3x3 of the values of x, y, z, look:

To calculate the determinant of a matrix, we will multiply all values of the main diagonal and subtract the secondary diagonal from it.

Let me explain in more detail in a drawing:

$\begin{gathered} Det=Main-(Secundary) \\ Det=16-(1) \\ Det=16-1 \\ Det=15 \end{gathered}$

$D=\begin{bmatrix}{1} & {-2} & {3} \\ {2} & {1} & {1} \\ {-3} & {2} & {-2}\end{bmatrix}=15$

Now, let's calculate three more determinants. This time, we will remove the column of the variable that we are working with and replace it with the column of the results, look:

$\begin{gathered} D_X=\begin{bmatrix}{7} & {-2} & {3} \\ {4} & {1} & {1} \\ {-10} & {2} & {-2}\end{bmatrix}=30 \\ \\ D_Y=\begin{bmatrix}{1} & 7 & {3} \\ 2 & 4 & {1} \\ -3 & -10 & {-2}\end{bmatrix}=-15 \\ \\ D_Z=\begin{bmatrix}{1} & {-2} & {7} \\ {2} & {1} & {4} \\ {-3} & {2} & {-10}\end{bmatrix}=15 \end{gathered}$

To calculate the values of x, y and z, we have to divide the determinant of the variable by the determinant of the initial matrix, look:

$\begin{gathered} x=\frac{D_X}{D}=\frac{30}{15}=2 \\ \\ \\ y=\frac{D_Y}{D}=\frac{-15}{15}=-1 \\ \\ \\ z=\frac{D_Z}{D}=\frac{15}{15}=1 \end{gathered}$

Answer:

(x, y, z) = (2, -1, 1)

X = 2

Y = -1

Z = 1