Question

How do you solve `9a - 4b = -5` and `6a - 2b = -3` using matrices?

Answer 1

`x=-0.33`
`y=0.5`

Explanation:

Look at the steps -

Answer 2

`:. a =-1/3 and b = 1/2`

Explanation:

Although the method might seem quite daunting, once the preparation process is mastered, the method itself is surprisingly quick and easy, involving a few simple calculations.
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We have the following equations:

`9a - 4b= -5 " and "6a - 2b = -3`

First write them as matrices:

`((9,-4),(6,-2)) ((a),(b)) = ((-5),(-3))`

Now find the inverse matrix of `A = ((9,-4),(6,-2))`

`abs(A) = (9xx -2)-(6xx-4) = -18+24 = 6`

`A^-1 = 1/6((-2,4),(-6,9)) = (color(red)((-1/3,2/3),(-1,3/2)))`
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Multiply both sides of the matrix equation by the inverse matrix.

`(color(red)((-1/3,2/3),(-1,3/2)))((9,-4),(6,-2)) ((a),(b)) = (color(red)((-1/3,2/3),(-1,3/2)))((-5),(-3))`

`color(white)(xxxxxxxxx)((1,0),(0,1)) ((a),(b)) = ((-1/3),(1/2))`

`color(white)(xxxxxxxxxxxxxx)((a),(b)) = ((-1/3),(1/2))`

`:. a =-1/3 and b = 1/2`

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Background knowledge... to help with the method above..
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A 2 x 2 matrix multiplied by the unit matrix remains unchanged

`((1,0),(0,1)) ((a,b),(c,d)) = ((a,b),(c,d))`

A matrix multiplied by its inverse gives the unit matrix -
also known as the Identity Matrix.

`A xx A^-1 = I = ((1,0),(0,1))`

To find the inverse matrix (`M^-1`) of matrix M

`M = ((a,b),(c,d))`

1. Find the determinant `(abs(M)) = ad-bc`

2. `M^-1 = 1/((abs(M)))( ( d,-b),(-c,a))`

(swop a and d and change the signs of b and c), then divide by the determinant.)