Linear Algebra Examples

$- 2 = x - 3 y$ , $2 = y - 2 x$

Step 1

Solve for $x$ in $- 2 = x - 3 y$ .

Tap for more steps...

Step 1.1

Rewrite the equation as $x - 3 y = - 2$ .

$x - 3 y = - 2$

$2 = y - 2 x$

Step 1.2

Add $3 y$ to both sides of the equation.

$x = - 2 + 3 y$

$2 = y - 2 x$

$x = - 2 + 3 y$

$2 = y - 2 x$

Step 2

Replace all occurrences of $x$ with $- 2 + 3 y$ in each equation.

Tap for more steps...

Step 2.1

Replace all occurrences of $x$ in $2 = y - 2 x$ with $- 2 + 3 y$ .

$2 = y - 2 (- 2 + 3 y)$

$x = - 2 + 3 y$

Step 2.2

Simplify the right side.

Tap for more steps...

Step 2.2.1

Simplify $y - 2 (- 2 + 3 y)$ .

Tap for more steps...

Step 2.2.1.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1.1

Apply the distributive property.

$2 = y - 2 \cdot - 2 - 2 (3 y)$

$x = - 2 + 3 y$

Step 2.2.1.1.2

Multiply $- 2$ by $- 2$ .

$2 = y + 4 - 2 (3 y)$

$x = - 2 + 3 y$

Step 2.2.1.1.3

Multiply $3$ by $- 2$ .

$2 = y + 4 - 6 y$

$x = - 2 + 3 y$

$2 = y + 4 - 6 y$

$x = - 2 + 3 y$

Step 2.2.1.2

Subtract $6 y$ from $y$ .

$2 = - 5 y + 4$

$x = - 2 + 3 y$

$2 = - 5 y + 4$

$x = - 2 + 3 y$

$2 = - 5 y + 4$

$x = - 2 + 3 y$

$2 = - 5 y + 4$

$x = - 2 + 3 y$

Step 3

Solve for $y$ in $2 = - 5 y + 4$ .

Tap for more steps...

Step 3.1

Rewrite the equation as $- 5 y + 4 = 2$ .

$- 5 y + 4 = 2$

$x = - 2 + 3 y$

Step 3.2

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

Tap for more steps...

Step 3.2.1

Subtract $4$ from both sides of the equation.

$- 5 y = 2 - 4$

$x = - 2 + 3 y$

Step 3.2.2

Subtract $4$ from $2$ .

$- 5 y = - 2$

$x = - 2 + 3 y$

$- 5 y = - 2$

$x = - 2 + 3 y$

Step 3.3

Divide each term in $- 5 y = - 2$ by $- 5$ and simplify.

Tap for more steps...

Step 3.3.1

Divide each term in $- 5 y = - 2$ by $- 5$ .

$\frac{- 5 y}{- 5} = \frac{- 2}{- 5}$

$x = - 2 + 3 y$

Step 3.3.2

Simplify the left side.

Tap for more steps...

Step 3.3.2.1

Cancel the common factor of $- 5$ .

Tap for more steps...

Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 5 y}{- 5} = \frac{- 2}{- 5}$

$x = - 2 + 3 y$

Step 3.3.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 2}{- 5}$

$x = - 2 + 3 y$

$y = \frac{- 2}{- 5}$

$x = - 2 + 3 y$

$y = \frac{- 2}{- 5}$

$x = - 2 + 3 y$

Step 3.3.3

Simplify the right side.

Tap for more steps...

Step 3.3.3.1

Dividing two negative values results in a positive value.

$y = \frac{2}{5}$

$x = - 2 + 3 y$

$y = \frac{2}{5}$

$x = - 2 + 3 y$

$y = \frac{2}{5}$

$x = - 2 + 3 y$

$y = \frac{2}{5}$

$x = - 2 + 3 y$

Step 4

Replace all occurrences of $y$ with $\frac{2}{5}$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $y$ in $x = - 2 + 3 y$ with $\frac{2}{5}$ .

$x = - 2 + 3 (\frac{2}{5})$

$y = \frac{2}{5}$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $- 2 + 3 (\frac{2}{5})$ .

Tap for more steps...

Step 4.2.1.1

Multiply $3 (\frac{2}{5})$ .

Tap for more steps...

Step 4.2.1.1.1

Combine $3$ and $\frac{2}{5}$ .

$x = - 2 + \frac{3 \cdot 2}{5}$

$y = \frac{2}{5}$

Step 4.2.1.1.2

Multiply $3$ by $2$ .

$x = - 2 + \frac{6}{5}$

$y = \frac{2}{5}$

$x = - 2 + \frac{6}{5}$

$y = \frac{2}{5}$

Step 4.2.1.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = - 2 \cdot \frac{5}{5} + \frac{6}{5}$

$y = \frac{2}{5}$

Step 4.2.1.3

Combine $- 2$ and $\frac{5}{5}$ .

$x = \frac{- 2 \cdot 5}{5} + \frac{6}{5}$

$y = \frac{2}{5}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 2 \cdot 5 + 6}{5}$

$y = \frac{2}{5}$

Step 4.2.1.5

Simplify the numerator.

Tap for more steps...

Step 4.2.1.5.1

Multiply $- 2$ by $5$ .

$x = \frac{- 10 + 6}{5}$

$y = \frac{2}{5}$

Step 4.2.1.5.2

Add $- 10$ and $6$ .

$x = \frac{- 4}{5}$

$y = \frac{2}{5}$

$x = \frac{- 4}{5}$

$y = \frac{2}{5}$

Step 4.2.1.6

Move the negative in front of the fraction.

$x = - \frac{4}{5}$

$y = \frac{2}{5}$

$x = - \frac{4}{5}$

$y = \frac{2}{5}$

$x = - \frac{4}{5}$

$y = \frac{2}{5}$

$x = - \frac{4}{5}$

$y = \frac{2}{5}$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- \frac{4}{5}, \frac{2}{5})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- \frac{4}{5}, \frac{2}{5})$

Equation Form:

$x = - \frac{4}{5}, y = \frac{2}{5}$

Step 7