Question

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`?

Answer 1

`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.