Name: Solution of Two Simultaneous Linear Equations
Uploaded: 2019-07-19
Duration: 9 min 55 s
Description: Two linear equations in 2 variables taken together are called simultaneous linear equations. The solution to these equations is the ordered pair which satisf...

Question

Use the product $[\begin{matrix} 1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4 \end{matrix}] [\begin{matrix} - 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2 \end{matrix}]$ to solve the system of equations:

$x + 3 z = 9$ , $- x + 2 y - 2 z = 4$ , $2 x - 3 y + 4 z = - 3$ .

Accepted Answer

Given product is $[\begin{matrix} 1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4 \end{matrix}] [\begin{matrix} - 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2 \end{matrix}]$ and $x + 3 z = 9$ , $- x + 2 y - 2 z = 4$ , $2 x - 3 y + 4 z = - 3$ .

Suppose $A = [\begin{matrix} 1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4 \end{matrix}]$ , $B = [\begin{matrix} - 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2 \end{matrix}]$

$A \times B = [\begin{matrix} 1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4 \end{matrix}] [\begin{matrix} - 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2 \end{matrix}]$

$= [\begin{matrix} - 2 - 9 + 12 & 0 - 2 + 2 & 1 + 3 - 4 \\ 0 + 18 - 18 & 0 + 4 - 3 & 0 - 6 + 6 \\ - 6 - 18 + 24 & 0 - 4 + 4 & 3 + 6 - 8 \end{matrix}]$

$= [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]$

Since, $A \times B = I$ , $∴ B = A^{- 1} . . . . . (1)$

Now, the given system of equations is

$x + 3 z = 9$

$- x + 2 y - 2 z = 4$

$2 x - 3 y + 4 z = - 3$

This can also be represented as, $[\begin{matrix} 1 & 0 & 3 \\ - 1 & 2 & - 2 \\ 2 & - 3 & 4 \end{matrix}] [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 9 \\ 4 \\ - 3 \end{matrix}]$

Here, we can observe that $[\begin{matrix} 1 & 0 & 3 \\ - 1 & 2 & - 2 \\ 2 & - 3 & 4 \end{matrix}] = A^{T}$

So, $A^{T} [\begin{matrix} x \\ y \\ z \end{matrix}] = [\begin{matrix} 9 \\ 4 \\ - 3 \end{matrix}]$

pre Multiply the above expression by ${(A^{T})}^{- 1}$

$[\begin{matrix} x \\ y \\ z \end{matrix}] = {(A^{T})}^{- 1} [\begin{matrix} 9 \\ 4 \\ - 3 \end{matrix}]$

$[\begin{matrix} x \\ y \\ z \end{matrix}] = {(A^{- 1})}^{T} [\begin{matrix} 9 \\ 4 \\ - 3 \end{matrix}]$

$[\begin{matrix} x \\ y \\ z \end{matrix}] = B^{T} [\begin{matrix} 9 \\ 4 \\ - 3 \end{matrix}]$ [Using $(1)$ ]

$[\begin{matrix} x \\ y \\ z \end{matrix}] = {[\begin{matrix} - 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2 \end{matrix}]}^{T} [\begin{matrix} 9 \\ 4 \\ - 3 \end{matrix}]$

$= [\begin{matrix} - 2 & 9 & 6 \\ 0 & 2 & 1 \\ 1 & - 3 & - 2 \end{matrix}] [\begin{matrix} 9 \\ 4 \\ - 3 \end{matrix}]$

$= [\begin{matrix} - 18 + 36 - 18 \\ 0 + 8 - 3 \\ 9 - 12 + 6 \end{matrix}]$

$= [\begin{matrix} 0 \\ 5 \\ 3 \end{matrix}]$

Hence, $x = 0, y = 5$ and $z = 3$ .

Use the product $[\begin{matrix} 1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4 \end{matrix}] [\begin{matrix} - 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2 \end{matrix}]$ to solve the system of equations:

$x + 3 z = 9$ , $- x + 2 y - 2 z = 4$ , $2 x - 3 y + 4 z = - 3$ .

Important Questions on Solution of Simultaneous Linear Equations

Learn and Prepare for any exam you want !

Use the product 1-1202-33-24-20192-361-2 to solve the system of equations: x+3z=9, −x+ 2y−2z=4, 2x−3y+4z=−3.

Important Questions on Solution of Simultaneous Linear Equations

Learn and Prepare for any exam you want !

Use the product $[\begin{matrix} 1 & - 1 & 2 \\ 0 & 2 & - 3 \\ 3 & - 2 & 4 \end{matrix}] [\begin{matrix} - 2 & 0 & 1 \\ 9 & 2 & - 3 \\ 6 & 1 & - 2 \end{matrix}]$ to solve the system of equations:

$x + 3 z = 9$ , $- x + 2 y - 2 z = 4$ , $2 x - 3 y + 4 z = - 3$ .