Answer to Question #297359 in Linear Algebra for Royal

Question

{F} Work Out

1. Solve by crammer's rule, the following equations.

3X + 3Y - Z = 11

2X - Y + 2Z = 9

4X + 3Y - 27 = 25

2. solve the above equations using graphical method. 

Answer 1

Solution:

1. Cramer's rule:

$\begin{aligned} &\Delta=\left|\begin{array}{ccc} 3 & 3 & -1 \\ 2 & -1 & 2 \\ 4 & 3 & 2 \end{array}\right| \\ &=3(-2-6)-3(4-8)-1(6+4) \\ &=3(-8)-3(-4)-1(10) \\ &=-24+12-10 \\ &=-22 \neq 0 \\ &\Delta_{\mathrm{X}}=\left|\begin{array}{ccc} 11 & 3 & -1 \\ 9 & -1 & 2 \\ 25 & 3 & 2 \end{array}\right| \\ &=11(-2-6)-3(18-50)-1(27+25) \\ &=11(-8)-3(32)-1(52) \\ &=-88+96-52 \\ &=-44 \\ \Delta_y=|&\begin{array}{lll} 3 & 11 & -1 \\ 2 & 9 & 2 \\ 4 & 25 & 2 \end{array} \mid \end{aligned}$

$\begin{aligned} &=3(18-50)-11(4-8)-1(50-36) \\ &=3(32)-11(4)-1(14) \\ &=-96+44-14 \\ &=-66 \\ &\begin{aligned} \Delta_{\mathrm{z}} &=\left|\begin{array}{lll} 3 & 3 & 11 \\ 2 & -1 & 9 \\ 4 & 3 & 25 \end{array}\right| \\ &=3(-25-27)-3(50-36)+11(6+4) \\ &=3(-52)-3(14)+11(10) \\ &=-156-42+110 \\ &=-88 \\ & \text { By Cramer's rule } \mathrm{x}=\frac{\Delta_{x}}{\Delta}=\frac{-44}{-22}=2 \\ & \quad \mathrm{y}=\frac{\Delta_{y}}{\Delta}=\frac{-66}{-22}=3 \\ &=\frac{-88}{\Delta}=\frac{-22}{-22}=4 \end{aligned} \\ &\therefore x=2,y=3, \mathrm{z}=4 \end{aligned}$

2. Graphical method:

We plot all the 3 equations in a 3d-system, where they intersect, that is our solution.

We find that they intersect at (2,3,4).