How do I use matrices to find the solution of the system of equations #y=1/3x+7/3# and #y=−5/4x+11/4#?

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Answer 1 · 2015-08-16T17:41:33

#color(red)(x=5/19,y=46/19)#

One way is to use Cramer's Rule.

Step 1. Enter your equations.

#y=1/3x+7/3#
#y=-5/4x+11/4#

Step 2. Write them in standard form.

#1/3x-y=-7/3#
#5/4x+y=11/4#

Step 3. Multiply to get rid of fractions.

#x-3y=-7#
#5x+4y=11#

The left hand side gives us the coefficient matrix.

#((1,-3),(5,4))#

The right hand side gives us the answer matrix.

#((-7),(11))#

The determinant #D# of the coefficient matrix is

#D = |(1,-3),(5,4)| = 4+15=19#

Let #D_x# be the determinant formed by replacing the #x#-column values with the answer-column values:

#D_x=|(-7,-3),(11,4)| = -28+33 = 5#

Similarly,

#D_y=|(1,-7),(5,11)|=11+35=46#

Cramer's Rule says that

#x = D_x/D =5/19#,

#y = D_y/D=46/19#,

The solution is #x=5/19,y=46/19#

Check:

#y=1/3x+7/3#

#46/19=1/3(5/19)+7/3 =5/57 +7/3 =(5/57+7×19/57) = (5+133)/57=138/57#

#46/19=46/19#

#y=-5/4x+11/4#

#46/19=-5/4×5/19+11/4=-25/78+(11×19)/78=(-25+209)/78=184/78#

#46/19=46/19#

It works!

Our solution is correct.