Linear Algebra Examples

$x + 3 y - 6 = 0$ , $- x - 3 y = - 6$

Step 1

Add $6$ to both sides of the equation.

$x + 3 y = 6, - x - 3 y = - 6$

Step 2

Write the system of equations in matrix form.

$[\begin{array}{c} 1 & 3 & 6 \\ - 1 & - 3 & - 6 \end{array}]$

Step 3

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

Tap for more steps...

Step 3.1

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

$[\begin{array}{c} 1 & 3 & 6 \\ - 1 + 1 \cdot 1 & - 3 + 1 \cdot 3 & - 6 + 1 \cdot 6 \end{array}]$

Step 3.2

Simplify $R_{2}$ .

$[\begin{array}{c} 1 & 3 & 6 \\ 0 & 0 & 0 \end{array}]$

Step 4

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

$x + 3 y = 6$

Step 5

Solve the equation for $y$ .

Tap for more steps...

Step 5.1

Subtract $x$ from both sides of the equation.

$3 y = 6 - x$

Step 5.2

Divide each term in $3 y = 6 - x$ by $3$ and simplify.

Tap for more steps...

Step 5.2.1

Divide each term in $3 y = 6 - x$ by $3$ .

$\frac{3 y}{3} = \frac{6}{3} + \frac{- x}{3}$

Step 5.2.2

Simplify the left side.

Tap for more steps...

Step 5.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 5.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{6}{3} + \frac{- x}{3}$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{6}{3} + \frac{- x}{3}$

Step 5.2.3

Simplify the right side.

Tap for more steps...

Step 5.2.3.1

Simplify each term.

Tap for more steps...

Step 5.2.3.1.1

Divide $6$ by $3$ .

$y = 2 + \frac{- x}{3}$

Step 5.2.3.1.2

Move the negative in front of the fraction.

$y = 2 - \frac{x}{3}$

Step 6

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

$(x, 2 - \frac{x}{3})$

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.

$X = [\begin{array}{c} x \\ y \end{array}] = [\begin{array}{c} x \\ 2 - \frac{x}{3} \end{array}]$