Question

# The cost of 4 dozen pencils, 3 dozen pens and 2 dozen erasers is Rs. 60. The cost of 2 dozen pencils, 4 dozen pens and 6 dozen erasers is Rs. 90 whereas the cost of 6 dozen pencils, 2 dozen pens and 3 dozen erasers is Rs. 70. Find the cost of each items per dozen by using matrices

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Solution

## Let the cost of one dozen pencils, one dozen pends and one dozen erasers be Rs.x,Rs.y and Rs.z respectively.According to the questions,4x+3y+2z=602x+4y+6z=906x+2y+3z=70The above equations can be expressed in the matrix form as⎡⎢⎣432246623⎤⎥⎦⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣609070⎤⎥⎦Using R2→R2−12R1 and R3→R3−32R1⎡⎢ ⎢ ⎢ ⎢ ⎢⎣43205250−520⎤⎥ ⎥ ⎥ ⎥ ⎥⎦⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣6060−20⎤⎥⎦Using R3→R3+R2, we get⎡⎢ ⎢ ⎢⎣4320525005⎤⎥ ⎥ ⎥⎦⎡⎢⎣xyz⎤⎥⎦=⎡⎢⎣606040⎤⎥⎦Now the equations can be rewrite in their original form as5z=40....(i)⇒z=852y+5z=60....(ii)4x+3y+2z=60....(iii)Putting the value of z in equation (ii),52y+5(8)=60y=(60−40)5×2=8⇒y=8Putting the values of y and z in equations (iii), we get4x+3(8)+2(8)=604x=60−24−16x=204=5∴x=5∴ The cost of each item per dozen are Rs.5,Rs.8 and Rs.8 respectively.

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