Linear Algebra Examples

$x + 3 y + z = a$ , $x - 2 y + z = b$ , $3 x + 7 y - z = c$

Step 1

Subtract $a$ from both sides of the equation.

$x + 3 y + z - a = 0, x - 2 y + z = b, 3 x + 7 y - z = c$

Step 2

Subtract $b$ from both sides of the equation.

$x + 3 y + z - a = 0, x - 2 y + z - b = 0, 3 x + 7 y - z = c$

Step 3

Subtract $c$ from both sides of the equation.

$x + 3 y + z - a = 0, x - 2 y + z - b = 0, 3 x + 7 y - z - c = 0$

Step 4

Write the system of equations in matrix form.

$[\begin{array}{c} - 1 & 0 & 0 & 1 & 3 & 1 & 0 \\ 0 & - 1 & 0 & 1 & - 2 & 1 & 0 \\ 0 & 0 & - 1 & 3 & 7 & - 1 & 0 \end{array}]$

Step 5

Find the reduced row echelon form.

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Step 5.1

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

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Step 5.1.1

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

$[\begin{array}{c} - - 1 & - 0 & - 0 & - 1 \cdot 1 & - 1 \cdot 3 & - 1 \cdot 1 & - 0 \\ 0 & - 1 & 0 & 1 & - 2 & 1 & 0 \\ 0 & 0 & - 1 & 3 & 7 & - 1 & 0 \end{array}]$

Step 5.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 0 & - 1 & - 3 & - 1 & 0 \\ 0 & - 1 & 0 & 1 & - 2 & 1 & 0 \\ 0 & 0 & - 1 & 3 & 7 & - 1 & 0 \end{array}]$

Step 5.2

Multiply each element of $R_{2}$ by $- 1$ to make the entry at $2, 2$ a $1$ .

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Step 5.2.1

Multiply each element of $R_{2}$ by $- 1$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 0 & 0 & - 1 & - 3 & - 1 & 0 \\ - 0 & - - 1 & - 0 & - 1 \cdot 1 & - - 2 & - 1 \cdot 1 & - 0 \\ 0 & 0 & - 1 & 3 & 7 & - 1 & 0 \end{array}]$

Step 5.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & 0 & - 1 & - 3 & - 1 & 0 \\ 0 & 1 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & - 1 & 3 & 7 & - 1 & 0 \end{array}]$

Step 5.3

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

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Step 5.3.1

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

$[\begin{array}{c} 1 & 0 & 0 & - 1 & - 3 & - 1 & 0 \\ 0 & 1 & 0 & - 1 & 2 & - 1 & 0 \\ - 0 & - 0 & - - 1 & - 1 \cdot 3 & - 1 \cdot 7 & - - 1 & - 0 \end{array}]$

Step 5.3.2

Simplify $R_{3}$ .

$[\begin{array}{c} 1 & 0 & 0 & - 1 & - 3 & - 1 & 0 \\ 0 & 1 & 0 & - 1 & 2 & - 1 & 0 \\ 0 & 0 & 1 & - 3 & - 7 & 1 & 0 \end{array}]$

Step 6

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

$a - x - 3 y - z = 0$

$b - x + 2 y - z = 0$

$c - 3 x - 7 y + z = 0$

Step 7

Move all terms not containing $a$ to the right side of the equation.

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Step 7.1

Add $x$ to both sides of the equation.

$a - 3 y - z = x$

$b - x + 2 y - z = 0$

$c - 3 x - 7 y + z = 0$

Step 7.2

Add $3 y$ to both sides of the equation.

$a - z = x + 3 y$

$b - x + 2 y - z = 0$

$c - 3 x - 7 y + z = 0$

Step 7.3

Add $z$ to both sides of the equation.

$a = x + 3 y + z$

$b - x + 2 y - z = 0$

$c - 3 x - 7 y + z = 0$

$a = x + 3 y + z$

$b - x + 2 y - z = 0$

$c - 3 x - 7 y + z = 0$

Step 8

Move all terms not containing $b$ to the right side of the equation.

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Step 8.1

Add $x$ to both sides of the equation.

$b + 2 y - z = x$

$a = x + 3 y + z$

$c - 3 x - 7 y + z = 0$

Step 8.2

Subtract $2 y$ from both sides of the equation.

$b - z = x - 2 y$

$a = x + 3 y + z$

$c - 3 x - 7 y + z = 0$

Step 8.3

Add $z$ to both sides of the equation.

$b = x - 2 y + z$

$a = x + 3 y + z$

$c - 3 x - 7 y + z = 0$

$b = x - 2 y + z$

$a = x + 3 y + z$

$c - 3 x - 7 y + z = 0$

Step 9

Move all terms not containing $c$ to the right side of the equation.

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Step 9.1

Add $3 x$ to both sides of the equation.

$c - 7 y + z = 3 x$

$a = x + 3 y + z$

$b = x - 2 y + z$

Step 9.2

Add $7 y$ to both sides of the equation.

$c + z = 3 x + 7 y$

$a = x + 3 y + z$

$b = x - 2 y + z$

Step 9.3

Subtract $z$ from both sides of the equation.

$c = 3 x + 7 y - z$

$a = x + 3 y + z$

$b = x - 2 y + z$

$c = 3 x + 7 y - z$

$a = x + 3 y + z$

$b = x - 2 y + z$

Step 10

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

$(x + 3 y + z, x - 2 y + z, 3 x + 7 y - z, x, y, z)$

Step 11

Decompose a solution vector by re-arranging each equation represented in the row-reduced form of the augmented matrix by solving for the dependent variable in each row yields the vector equality.

$X = [\begin{array}{c} a \\ b \\ c \\ x \\ y \\ z \end{array}] = [\begin{array}{c} x + 3 y + z \\ x - 2 y + z \\ 3 x + 7 y - z \\ x \\ y \\ z \end{array}]$