Linear Algebra Examples

$y = - x + 4$ , $x + y = 4$

Step 1

Add $x$ to both sides of the equation.

$y + x = 4, x + y = 4$

Step 2

Write the system of equations in matrix form.

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

Step 3

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

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Step 3.1

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

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

Step 3.2

Simplify $R_{2}$ .

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

Step 4

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

$x + y = 4$

Step 5

Subtract $x$ from both sides of the equation.

$y = 4 - x$

Step 6

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

$(x, 4 - x)$

Step 7

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.

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