Question

Solve the system by using elementary row operations on the equations. Follow the systematic elimination procedure

x1+4x2 =11
2x1+7x2=18
Find the solution to the system of equations.

Answer 1

The solution of the Given matrix

( x₁ , x ₂ ) = ( - 5 , 4 )

Explanation:

Step(i):-

Given equations are x₁+4 x₂ = 11 ...(i)

2 x₁ + 7 x₂= 18 ...(ii)

The matrix form

A X = B

`\left[\begin{array}{ccc}1&4\\2&7\\\end{array}\right] \left[\begin{array}{ccc}x\\y\\\end{array}\right] = \left[\begin{array}{ccc}11\\8\\\end{array}\right]`

Step(ii):-

`\left[\begin{array}{ccc}1&4\\2&7\\\end{array}\right] \left[\begin{array}{ccc}x\\y\\\end{array}\right] = \left[\begin{array}{ccc}11\\8\\\end{array}\right]`

The Augmented Matrix form is

`[AB] = \left[\begin{array}{ccc}1&4&11\\2&7&18\\\end{array}\right]`

Apply Row operations, R₂ → R₂-2 R₁

`[AB] = \left[\begin{array}{ccc}1&4&11\\0&-1&-4\\\end{array}\right]`

The matrix form

`\left[\begin{array}{ccc}1&4\\0&-1\\\end{array}\right] \left[\begin{array}{ccc}x\\y\\\end{array}\right] = \left[\begin{array}{ccc}11\\-4\\\end{array}\right]`

The equations are

x₁ + 4 x₂ = 11 ...(a)

- x ₂ = - 4

x ₂ = 4

Substitute x ₂ = 4 in equation (a)

x₁ + 4 x₂ = 11

x₁ = 11 - 16

x₁ = -5

Final answer:-

The solution of the Given matrix

( x₁ , x ₂ ) = ( - 5 , 4 )