Question

use the matrix tool to solve the system of equations choose the correct ordered pair4x-y=10 and 8x+5y=34

Answer 1

Answer:

  $\left[\begin{array}{ccc}x\\y\\\end{array}\right] = \left[\begin{array}{ccc}3\\2\\\end{array}\right]$

Solution of the given system of equations

 x = 3 and  y= 2

                     

Step-by-step explanation:

Step(i):-

Given system of equations are

                                      4x - y =  10 ...(i)

                                     8x +5y = 34...(ii)

The matrix form

                          A X = B

$\left[\begin{array}{ccc}4&-1\\8&5\\\end{array}\right] \left[\begin{array}{ccc}x\\y\\\end{array}\right] = \left[\begin{array}{ccc}10\\34\\\end{array}\right]$

Step(ii):-

By using matrix inversion method

$A^{-1} = \frac{1}{|A|} adj A$

|A| = ad-b c = 4(5) - 8(-1) = 20+8 = 28

Given matrix

            $A = \left[\begin{array}{ccc}4&-1\\8&5\\\end{array}\right]$

$AdjA = \left[\begin{array}{ccc}d&-b\\-c&a\\\end{array}\right]$

$A djA = \left[\begin{array}{ccc}5&1\\-8&4\\\end{array}\right]$

Inverse of the matrix

                           $A^{-1} = \frac{1}{|A|} adj A$

                          $A^{-1} = \frac{1}{|A|} A djA =\frac{1}{28} \left[\begin{array}{ccc}5&1\\-8&4\\\end{array}\right]$

Step(iii):-

The solution of the given system of equations by using matrix inversion method

                     X = A⁻¹ B

                    $X = A^{-1}B =\frac{1}{28} \left[\begin{array}{ccc}5&1\\-8&4\\\end{array}\right]\left[\begin{array}{ccc}10\\34\\\end{array}\right]$

                   $X = A^{-1}B =\frac{1}{28} \left[\begin{array}{ccc}5X10+1X34\\-8X10+4X34\\\end{array}\right]$

                 $\left[\begin{array}{ccc}x\\y\\\end{array}\right] =\frac{1}{28} \left[\begin{array}{ccc}84\\56\\\end{array}\right]$

                 $\left[\begin{array}{ccc}x\\y\\\end{array}\right] = \left[\begin{array}{ccc}3\\2\\\end{array}\right]$

Final answer:-

Solution of the given system   x = 3 and  y= 2