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

2 Answers
Sep 22, 2016

Answer:

#x=-0.33#
#y=0.5#

Explanation:

Look at the steps -

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Sep 22, 2016

Answer:

# :. 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.)