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?

1 Answer
Aug 16, 2015

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

Explanation:

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.