Linear Algebra Examples

$1 x + 2 y = 4$ , $3 x + 36 = 48$

Step 1

Multiply $x$ by $1$ .

$x + 2 y = 4, 3 x + 36 = 48$

Step 2

Subtract $36$ from both sides of the equation.

$x + 2 y = 4, 3 x = 48 - 36$

Step 3

Subtract $36$ from $48$ .

$x + 2 y = 4, 3 x = 12$

Step 4

Write the system of equations in matrix form.

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

Step 5

Find the reduced row echelon form.

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Step 5.1

Perform the row operation $R_{2} = R_{2} - 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

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Step 5.1.1

Perform the row operation $R_{2} = R_{2} - 3 R_{1}$ to make the entry at $2, 1$ a $0$ .

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

Step 5.1.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 4 \\ 0 & - 6 & 0 \end{array}]$

Step 5.2

Multiply each element of $R_{2}$ by $- \frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

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Step 5.2.1

Multiply each element of $R_{2}$ by $- \frac{1}{6}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & 2 & 4 \\ - \frac{1}{6} \cdot 0 & - \frac{1}{6} \cdot - 6 & - \frac{1}{6} \cdot 0 \end{array}]$

Step 5.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 2 & 4 \\ 0 & 1 & 0 \end{array}]$

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

Simplify $R_{1}$ .

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

Step 6

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

$x = 4$

$y = 0$

Step 7

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

$(4, 0)$

Step 8

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.

$X = [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 4 \\ 0 \end{array}]$