Linear Algebra Examples

$(2 x + y) = a + 9$ , $(x - 2 y) = b - 2$

Step 1

Remove parentheses.

$2 x + y = a + 9, x - 2 y = b - 2$

Step 2

Subtract $a$ from both sides of the equation.

$2 x + y - a = 9, x - 2 y = b - 2$

Step 3

Subtract $b$ from both sides of the equation.

$2 x + y - a = 9, x - 2 y - b = - 2$

Step 4

Write the system of equations in matrix form.

$[\begin{array}{c} - 1 & 0 & 2 & 1 & 9 \\ 0 & - 1 & 1 & - 2 & - 2 \end{array}]$

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 $1$ .

$[\begin{array}{c} - - 1 & - 0 & - 1 \cdot 2 & - 1 \cdot 1 & - 1 \cdot 9 \\ 0 & - 1 & 1 & - 2 & - 2 \end{array}]$

Step 5.1.2

Simplify $R_{1}$ .

$[\begin{array}{c} 1 & 0 & - 2 & - 1 & - 9 \\ 0 & - 1 & 1 & - 2 & - 2 \end{array}]$

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 $1$ .

$[\begin{array}{c} 1 & 0 & - 2 & - 1 & - 9 \\ - 0 & - - 1 & - 1 \cdot 1 & - - 2 & - - 2 \end{array}]$

Step 5.2.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 0 & - 2 & - 1 & - 9 \\ 0 & 1 & - 1 & 2 & 2 \end{array}]$

Step 6

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

$a - 2 x - y = - 9$

$b - x + 2 y = 2$

Step 7

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

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Step 7.1

Add $2 x$ to both sides of the equation.

$a - y = - 9 + 2 x$

$b - x + 2 y = 2$

Step 7.2

Add $y$ to both sides of the equation.

$a = - 9 + 2 x + y$

$b - x + 2 y = 2$

$a = - 9 + 2 x + y$

$b - x + 2 y = 2$

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.

$b + 2 y = 2 + x$

$a = - 9 + 2 x + y$

Step 8.2

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

$b = 2 + x - 2 y$

$a = - 9 + 2 x + y$

$b = 2 + x - 2 y$

$a = - 9 + 2 x + y$

Step 9

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

$(- 9 + 2 x + y, 2 + x - 2 y, x, y)$

Step 10

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.

$X = [\begin{array}{c} a \\ b \\ x \\ y \end{array}] = [\begin{array}{c} - 9 + 2 x + y \\ 2 + x - 2 y \\ x \\ y \end{array}]$