Linear Algebra Examples

$x + 2 y = 48$ , $x + y = 30$ , $2 x + y = 50$ , $f = 70 x + 30 y$

Step 1

Move all terms containing variables to the left side of the equation.

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Step 1.1

Subtract $70 x$ from both sides of the equation.

$x + 2 y = 48, x + y = 30, 2 x + y = 50, f - 70 x = 30 y$

Step 1.2

Subtract $30 y$ from both sides of the equation.

$x + 2 y = 48, x + y = 30, 2 x + y = 50, f - 70 x - 30 y = 0$

Step 2

Write the system of equations in matrix form.

$[\begin{array}{c} 0 & 1 & 2 & 48 \\ 0 & 1 & 1 & 30 \\ 0 & 2 & 1 & 50 \\ 1 & - 70 & - 30 & 0 \end{array}]$

Step 3

Find the reduced row echelon form.

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Step 3.1

Swap $R_{4}$ with $R_{1}$ to put a nonzero entry at $1, 1$ .

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 30 \\ 0 & 2 & 1 & 50 \\ 0 & 1 & 2 & 48 \end{array}]$

Step 3.2

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

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Step 3.2.1

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

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 30 \\ 0 - 2 \cdot 0 & 2 - 2 \cdot 1 & 1 - 2 \cdot 1 & 50 - 2 \cdot 30 \\ 0 & 1 & 2 & 48 \end{array}]$

Step 3.2.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 30 \\ 0 & 0 & - 1 & - 10 \\ 0 & 1 & 2 & 48 \end{array}]$

Step 3.3

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

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Step 3.3.1

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

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 30 \\ 0 & 0 & - 1 & - 10 \\ 0 - 0 & 1 - 1 & 2 - 1 & 48 - 30 \end{array}]$

Step 3.3.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 30 \\ 0 & 0 & - 1 & - 10 \\ 0 & 0 & 1 & 18 \end{array}]$

Step 3.4

Multiply each element of $R_{3}$ by $- 1$ to make the entry at $3, 3$ a $1$ .

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Step 3.4.1

Multiply each element of $R_{3}$ by $- 1$ to make the entry at $3, 3$ a $1$ .

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 30 \\ - 0 & - 0 & - - 1 & - - 10 \\ 0 & 0 & 1 & 18 \end{array}]$

Step 3.4.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 30 \\ 0 & 0 & 1 & 10 \\ 0 & 0 & 1 & 18 \end{array}]$

Step 3.5

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

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Step 3.5.1

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

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 30 \\ 0 & 0 & 1 & 10 \\ 0 - 0 & 0 - 0 & 1 - 1 & 18 - 10 \end{array}]$

Step 3.5.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 30 \\ 0 & 0 & 1 & 10 \\ 0 & 0 & 0 & 8 \end{array}]$

Step 3.6

Multiply each element of $R_{4}$ by $\frac{1}{8}$ to make the entry at $4, 4$ a $1$ .

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Step 3.6.1

Multiply each element of $R_{4}$ by $\frac{1}{8}$ to make the entry at $4, 4$ a $1$ .

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 30 \\ 0 & 0 & 1 & 10 \\ \frac{0}{8} & \frac{0}{8} & \frac{0}{8} & \frac{8}{8} \end{array}]$

Step 3.6.2

Simplify $R_{4}$ .

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 30 \\ 0 & 0 & 1 & 10 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 3.7

Perform the row operation $R_{3} = R_{3} - 10 R_{4}$ to make the entry at $3, 4$ a $0$ .

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Step 3.7.1

Perform the row operation $R_{3} = R_{3} - 10 R_{4}$ to make the entry at $3, 4$ a $0$ .

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 30 \\ 0 - 10 \cdot 0 & 0 - 10 \cdot 0 & 1 - 10 \cdot 0 & 10 - 10 \cdot 1 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 3.7.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 30 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 3.8

Perform the row operation $R_{2} = R_{2} - 30 R_{4}$ to make the entry at $2, 4$ a $0$ .

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Step 3.8.1

Perform the row operation $R_{2} = R_{2} - 30 R_{4}$ to make the entry at $2, 4$ a $0$ .

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 - 30 \cdot 0 & 1 - 30 \cdot 0 & 1 - 30 \cdot 0 & 30 - 30 \cdot 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 3.8.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 3.9

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

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Step 3.9.1

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

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 - 0 & 1 - 0 & 1 - 1 & 0 - 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 3.9.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 70 & - 30 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 3.10

Perform the row operation $R_{1} = R_{1} + 30 R_{3}$ to make the entry at $1, 3$ a $0$ .

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Step 3.10.1

Perform the row operation $R_{1} = R_{1} + 30 R_{3}$ to make the entry at $1, 3$ a $0$ .

$[\begin{array}{c} 1 + 30 \cdot 0 & - 70 + 30 \cdot 0 & - 30 + 30 \cdot 1 & 0 + 30 \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 3.10.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 70 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 3.11

Perform the row operation $R_{1} = R_{1} + 70 R_{2}$ to make the entry at $1, 2$ a $0$ .

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Step 3.11.1

Perform the row operation $R_{1} = R_{1} + 70 R_{2}$ to make the entry at $1, 2$ a $0$ .

$[\begin{array}{c} 1 + 70 \cdot 0 & - 70 + 70 \cdot 1 & 0 + 70 \cdot 0 & 0 + 70 \cdot 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}]$

Step 3.11.2

Simplify $R_{1}$ .

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

Step 4

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

$f = 0$

$x = 0$

$y = 0$

$0 = 1$

Step 5

Since $0 \neq 1$ , there are no solutions.

No solution