Linear Algebra Examples

$x + 4 y = 4$ $2 x + 9 y = 11$ $5 x + 24 y = 32$

Step 1

Write the system as a matrix.

$[\begin{array}{c} 1 & 4 & 4 \\ 2 & 9 & 11 \\ 5 & 24 & 32 \end{array}]$

Step 2

Find the reduced row echelon form.

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Step 2.1

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

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Step 2.1.1

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

$[\begin{array}{c} 1 & 4 & 4 \\ 2 - 2 \cdot 1 & 9 - 2 \cdot 4 & 11 - 2 \cdot 4 \\ 5 & 24 & 32 \end{array}]$

Step 2.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 4 & 4 \\ 0 & 1 & 3 \\ 5 & 24 & 32 \end{array}]$

Step 2.2

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

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Step 2.2.1

Perform the row operation $R_{3} = R_{3} - 5 R_{1}$ to make the entry at $3, 1$ a $0$ .

$[\begin{array}{c} 1 & 4 & 4 \\ 0 & 1 & 3 \\ 5 - 5 \cdot 1 & 24 - 5 \cdot 4 & 32 - 5 \cdot 4 \end{array}]$

Step 2.2.2

Simplify $R_{3}$ .

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

Step 2.3

Perform the row operation $R_{3} = R_{3} - 4 R_{2}$ to make the entry at $3, 2$ a $0$ .

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Step 2.3.1

Perform the row operation $R_{3} = R_{3} - 4 R_{2}$ to make the entry at $3, 2$ a $0$ .

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

Step 2.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 4 & 4 \\ 0 & 1 & 3 \\ 0 & 0 & 0 \end{array}]$

Step 2.4

Perform the row operation $R_{1} = R_{1} - 4 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 2.4.1

Perform the row operation $R_{1} = R_{1} - 4 R_{2}$ to make the entry at $1, 2$ a $0$ .

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

Step 2.4.2

Simplify $R_{1}$ .

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

Step 3

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

$x = - 8$

$y = 3$

$0 = 0$

Step 4

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

$(- 8, 3)$