We will first rewrite the system by substituting

m

and

n

for

\frac{1}{x}

and

\frac{1}{y},

respectively.

{\frac{4}{x} + \frac{1}{y} = 1 \frac{8}{x} + \frac{4}{y} = 3 (I) (II)

$(I), (II):$ Write fraction as a mixed number

{4 \frac{1}{x} + \frac{1}{y} = 1 8 \frac{1}{x} + 4 \frac{1}{y} = 3

$(I), (II):$ $\frac{1}{x} = m$ , $\frac{1}{y} = n$

{4 m + n = 1 8 m + 4 n = 3

Now we can write the system as a matrix.

To rewrite the system of equations as a matrix, we need to consider how the elements of the system relate to the elements of a matrix.

The equals signs in the system of equations are represented with a vertical bar in the matrix.
The coefficients of the variables form the columns on the left-hand side of the bar.
The constants form the column on the right-hand side of the bar.

Below we demonstrate this in a generalized form.

{a_{11} m + a_{12} n = c_{1} a_{21} m + a_{22} n = c_{2} ⇕ [a_{11} a_{21} a_{12} a_{22} c_{1} c_{2}]

When each equation in the system is written in the same order, we can consider the coefficients of the variables and the constants.

{4 m + 1 n = 1 8 m + 4 n = 3

Now that we have identified all of the variables and constants, we can place them in a matrix.

[481413]

In order to solve the matrix, we will use row operations to obtain a matrix in the following form.

[1001 a b]

This final matrix represents the solution of the system of equations, where

m = a

and

n = b .

Let's solve the matrix!

[481413]

$(I):$ $LHS \cdot 2 = RHS \cdot 2$

[882423]

$(II):$ Subtract $(I)$

[8 8 - 8 2 4 - 2 2 3 - 2]

$(II):$ Subtract terms

[802221]

$(I):$ Subtract $(II)$

[8 - 0 0 2 - 2 2 2 - 1 1]

$(I):$ Subtract terms

[800211]

$(I):$ $LHS / 8 = RHS / 8$

⎣ ⎢ ⎢ ⎡ 1002 \frac{1}{8} 1 ⎦ ⎥ ⎥ ⎤

$(II):$ $LHS / 2 = RHS / 2$

⎣ ⎢ ⎢ ⎢ ⎡ 1001 \frac{1}{8} \frac{1}{2} ⎦ ⎥ ⎥ ⎥ ⎤

We have found that

m = \frac{1}{8}

and

n = \frac{1}{2} .

Now we can find

x

and

y .

m = \frac{1}{x} = \frac{1}{8} n = \frac{1}{y} = \frac{1}{2} \Rightarrow x = 8 \Rightarrow y = 2

The solution of the system is the point

(8, 2) .

Exercise