Linear Algebra Examples

$6.5 x - 6.5 y = 338$ , $x + 32.4 y = 338$

Step 1

Write the system of equations in matrix form.

$[\begin{array}{c} 6.5 & - 6.5 & 338 \\ 1 & 32.4 & 338 \end{array}]$

Step 2

Find the reduced row echelon form.

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Step 2.1

Multiply each element of $R_{1}$ by $\frac{1}{6.5}$ to make the entry at $1, 1$ a $1$ .

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Step 2.1.1

Multiply each element of $R_{1}$ by $\frac{1}{6.5}$ to make the entry at $1, 1$ a $1$ .

$[\begin{array}{c} \frac{6.5}{6.5} & \frac{- 6.5}{6.5} & \frac{338}{6.5} \\ 1 & 32.4 & 338 \end{array}]$

Step 2.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & - 1 & 52 \\ 1 & 32.4 & 338 \end{array}]$

Step 2.2

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

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Step 2.2.1

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

$[\begin{array}{c} 1 & - 1 & 52 \\ 1 - 1 & 32.4 + 1 & 338 - 52 \end{array}]$

Step 2.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 52 \\ 0 & 33.4 & 286 \end{array}]$

Step 2.3

Multiply each element of $R_{2}$ by $\frac{1}{33.4}$ to make the entry at $2, 2$ a $1$ .

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Step 2.3.1

Multiply each element of $R_{2}$ by $\frac{1}{33.4}$ to make the entry at $2, 2$ a $1$ .

$[\begin{array}{c} 1 & - 1 & 52 \\ \frac{0}{33.4} & \frac{33.4}{33.4} & \frac{286}{33.4} \end{array}]$

Step 2.3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & - 1 & 52 \\ 0 & 1 & 8.56287425 \end{array}]$

Step 2.4

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

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Step 2.4.1

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

$[\begin{array}{c} 1 + 0 & - 1 + 1 \cdot 1 & 52 + 1 \cdot 8.56287425 \\ 0 & 1 & 8.56287425 \end{array}]$

Step 2.4.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & 60.56287425 \\ 0 & 1 & 8.56287425 \end{array}]$

Step 3

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

$x = 60.56287425$

$y = 8.56287425$

Step 4

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

$(60.56287425, 8.56287425)$

Step 5

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.

$X = [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} 60.56287425 \\ 8.56287425 \end{array}]$