Finite Math Examples

$x - 2 y = 3$ , $8 x - 16 y = 24$ , $4 x - 8 y = 12$

Step 1

Write the system as a matrix.

$[\begin{array}{c} 1 & - 2 & 3 \\ 8 & - 16 & 24 \\ 4 & - 8 & 12 \end{array}]$

Step 2

Find the reduced row echelon form.

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Step 2.1

Perform the row operation $R_{2} = R_{2} - 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 2.1.1

Perform the row operation $R_{2} = R_{2} - 8 R_{1}$ to make the entry at $2, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 \\ 8 - 8 \cdot 1 & - 16 - 8 \cdot - 2 & 24 - 8 \cdot 3 \\ 4 & - 8 & 12 \end{array}]$

Step 2.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 2 & 3 \\ 0 & 0 & 0 \\ 4 & - 8 & 12 \end{array}]$

Step 2.2

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 2.2.1

Perform the row operation $R_{3} = R_{3} - 4 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & - 2 & 3 \\ 0 & 0 & 0 \\ 4 - 4 \cdot 1 & - 8 - 4 \cdot - 2 & 12 - 4 \cdot 3 \end{array}]$

Step 2.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]$

Step 3

Use the result matrix to declare the final solution to the system of equations.

$x - 2 y = 3$

$0 = 0$

Step 4

The solution is the set of ordered pairs that make the system true.

$(3 + 2 y, y)$