Linear Algebra Examples

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} x \\ y \end{array}] \cdot ([\begin{array}{c} 0 & 1 \\ - 3 & 1 \end{array}] [\begin{array}{c} x \\ y \end{array}])$

Step 1

Simplify.

Tap for more steps...

Step 1.1

Rewrite using the commutative property of multiplication.

$[\begin{array}{c} 21 \\ - 9 \end{array}] = 1 [\begin{array}{c} 0 & 1 \\ - 3 & 1 \end{array}] \cdot 1 [\begin{array}{c} x \\ y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.2

Multiply $1$ by each element of the matrix.

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} 1 \cdot 0 & 1 \cdot 1 \\ 1 \cdot - 3 & 1 \cdot 1 \end{array}] \cdot 1 [\begin{array}{c} x \\ y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.3

Simplify each element in the matrix.

Tap for more steps...

Step 1.3.1

Multiply $0$ by $1$ .

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} 0 & 1 \cdot 1 \\ 1 \cdot - 3 & 1 \cdot 1 \end{array}] \cdot 1 [\begin{array}{c} x \\ y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.3.2

Multiply $1$ by $1$ .

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} 0 & 1 \\ 1 \cdot - 3 & 1 \cdot 1 \end{array}] \cdot 1 [\begin{array}{c} x \\ y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.3.3

Multiply $- 3$ by $1$ .

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} 0 & 1 \\ - 3 & 1 \cdot 1 \end{array}] \cdot 1 [\begin{array}{c} x \\ y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.3.4

Multiply $1$ by $1$ .

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} 0 & 1 \\ - 3 & 1 \end{array}] \cdot 1 [\begin{array}{c} x \\ y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.4

Move $1$ to the left of $[\begin{array}{c} 0 & 1 \\ - 3 & 1 \end{array}]$ .

$[\begin{array}{c} 21 \\ - 9 \end{array}] = 1 \cdot [\begin{array}{c} 0 & 1 \\ - 3 & 1 \end{array}] [\begin{array}{c} x \\ y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.5

Multiply $1$ by each element of the matrix.

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} 1 \cdot 0 & 1 \cdot 1 \\ 1 \cdot - 3 & 1 \cdot 1 \end{array}] [\begin{array}{c} x \\ y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.6

Simplify each element in the matrix.

Tap for more steps...

Step 1.6.1

Multiply $0$ by $1$ .

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} 0 & 1 \cdot 1 \\ 1 \cdot - 3 & 1 \cdot 1 \end{array}] [\begin{array}{c} x \\ y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.6.2

Multiply $1$ by $1$ .

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} 0 & 1 \\ 1 \cdot - 3 & 1 \cdot 1 \end{array}] [\begin{array}{c} x \\ y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.6.3

Multiply $- 3$ by $1$ .

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} 0 & 1 \\ - 3 & 1 \cdot 1 \end{array}] [\begin{array}{c} x \\ y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.6.4

Multiply $1$ by $1$ .

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} 0 & 1 \\ - 3 & 1 \end{array}] [\begin{array}{c} x \\ y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.7

Multiply $[\begin{array}{c} 0 & 1 \\ - 3 & 1 \end{array}] [\begin{array}{c} x \\ y \end{array}]$ .

Tap for more steps...

Step 1.7.1

Two matrices can be multiplied if and only if the number of columns in the first matrix is equal to the number of rows in the second matrix. In this case, the first matrix is $2 \times 2$ and the second matrix is $2 \times 1$ .

Step 1.7.2

Multiply each row in the first matrix by each column in the second matrix.

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} 0 x + 1 y \\ - 3 x + 1 y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.7.3

Simplify each element of the matrix by multiplying out all the expressions.

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} y \\ - 3 x + y \end{array}] (1 [\begin{array}{c} x \\ y \end{array}])$

Step 1.8

Multiply $1$ by each element of the matrix.

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} y \\ - 3 x + y \end{array}] [\begin{array}{c} 1 x \\ 1 y \end{array}]$

Step 1.9

Simplify each element in the matrix.

Tap for more steps...

Step 1.9.1

Multiply $x$ by $1$ .

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} y \\ - 3 x + y \end{array}] [\begin{array}{c} x \\ 1 y \end{array}]$

Step 1.9.2

Multiply $y$ by $1$ .

$[\begin{array}{c} 21 \\ - 9 \end{array}] = [\begin{array}{c} y \\ - 3 x + y \end{array}] [\begin{array}{c} x \\ y \end{array}]$

Step 2

The matrix equation can be written as a set of equations.

$21 = y$

$- 9 = - 3 x + y$

Step 3

Rewrite the equation as $y = 21$ .

$y = 21$

$- 9 = - 3 x + y$

Step 4

Replace all occurrences of $y$ with $21$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $y$ in $- 9 = - 3 x + y$ with $21$ .

$- 9 = - 3 x + 21$

$y = 21$

Step 4.2

Simplify the left side.

Tap for more steps...

Step 4.2.1

Remove parentheses.

$- 9 = - 3 x + 21$

$y = 21$

$- 9 = - 3 x + 21$

$y = 21$

$- 9 = - 3 x + 21$

$y = 21$

Step 5

Solve for $x$ in $- 9 = - 3 x + 21$ .

Tap for more steps...

Step 5.1

Rewrite the equation as $- 3 x + 21 = - 9$ .

$- 3 x + 21 = - 9$

$y = 21$

Step 5.2

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

Tap for more steps...

Step 5.2.1

Subtract $21$ from both sides of the equation.

$- 3 x = - 9 - 21$

$y = 21$

Step 5.2.2

Subtract $21$ from $- 9$ .

$- 3 x = - 30$

$y = 21$

$- 3 x = - 30$

$y = 21$

Step 5.3

Divide each term in $- 3 x = - 30$ by $- 3$ and simplify.

Tap for more steps...

Step 5.3.1

Divide each term in $- 3 x = - 30$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 30}{- 3}$

$y = 21$

Step 5.3.2

Simplify the left side.

Tap for more steps...

Step 5.3.2.1

Cancel the common factor of $- 3$ .

Tap for more steps...

Step 5.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 30}{- 3}$

$y = 21$

Step 5.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 30}{- 3}$

$y = 21$

$x = \frac{- 30}{- 3}$

$y = 21$

$x = \frac{- 30}{- 3}$

$y = 21$

Step 5.3.3

Simplify the right side.

Tap for more steps...

Step 5.3.3.1

Divide $- 30$ by $- 3$ .

$x = 10$

$y = 21$

$x = 10$

$y = 21$

$x = 10$

$y = 21$

$x = 10$

$y = 21$

Step 6

Solve the system of equations.

$x = 10 y = 21$

Step 7

List all of the solutions.

$x = 10, y = 21$