Question

Using matrices, solve the following system of equations: x + 2y - 3z = - 4
2x + 3y + 2z = 2
3x - 3y – 4z = 11

Answer 1

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Answer 2

Given:

system of equations:

x + 2y - 3z = - 4

2x + 3y + 2z = 2

3x - 3y – 4z = 11

To Solve:

The system of equations using matrices.

Solution:

We can represent this in the matrix form.

• $\left[\begin{array}{ccc}1&2&-3\\2&3&2\\3&-3&-4\end{array}\right]$  $\left[\begin{array}{ccc}x\\y\\z\end{array}\right]$= $\left[\begin{array}{ccc}-4\\2\\11\end{array}\right]$

We can do the following steps to solve this equations.

1. Row 3 = Row 3 - 3 x Row 1
2. Row 2 = Row 2 - 2 x Row 1

• $\left[\begin{array}{ccc}1&2&-3\\0&-1&8\\0&-9&5\end{array}\right]$ $\left[\begin{array}{ccc}x\\y\\z\end{array}\right]$ = $\left[\begin{array}{ccc}-4\\10\\23\end{array}\right]$

Now do the next steps as :

1. Row 3 = Row3 - 9x Row2

• $\left[\begin{array}{ccc}1&2&-3\\0&-1&8\\0&0&-67\end{array}\right]$ $\left[\begin{array}{ccc}x\\y\\z\end{array}\right]$= $\left[\begin{array}{ccc}-4\\10\\-67\end{array}\right]$

Therefore,

In row 3

• -67z = -67 ==>
• z = 1

Applying in row 2,

• -y + 8 = 10
• y = -2

And in row 1,

• x -4 -3 = -4
• x = 3

Therefore,

Solution ofthe following system of equations: x + 2y - 3z = - 4

2x + 3y + 2z = 2 , 3x - 3y – 4z = 11 , is $\left[\begin{array}{ccc}x\\y\\z\end{array}\right]$  = $\left[\begin{array}{ccc}3\\-2\\1\end{array}\right]$.