Linear Algebra Examples

$x - 2 y = - 1$ , $2 x + y = 4$

Step 1

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

$x = - 1 + 2 y$

$2 x + y = 4$

Step 2

Replace all occurrences of $x$ with $- 1 + 2 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $2 x + y = 4$ with $- 1 + 2 y$ .

$2 (- 1 + 2 y) + y = 4$

$x = - 1 + 2 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $2 (- 1 + 2 y) + y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$2 \cdot - 1 + 2 (2 y) + y = 4$

$x = - 1 + 2 y$

Step 2.2.1.1.2

Multiply $2$ by $- 1$ .

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

$x = - 1 + 2 y$

Step 2.2.1.1.3

Multiply $2$ by $2$ .

$- 2 + 4 y + y = 4$

$x = - 1 + 2 y$

$- 2 + 4 y + y = 4$

$x = - 1 + 2 y$

Step 2.2.1.2

Add $4 y$ and $y$ .

$- 2 + 5 y = 4$

$x = - 1 + 2 y$

$- 2 + 5 y = 4$

$x = - 1 + 2 y$

$- 2 + 5 y = 4$

$x = - 1 + 2 y$

$- 2 + 5 y = 4$

$x = - 1 + 2 y$

Step 3

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

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Step 3.1

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

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Step 3.1.1

Add $2$ to both sides of the equation.

$5 y = 4 + 2$

$x = - 1 + 2 y$

Step 3.1.2

Add $4$ and $2$ .

$5 y = 6$

$x = - 1 + 2 y$

$5 y = 6$

$x = - 1 + 2 y$

Step 3.2

Divide each term in $5 y = 6$ by $5$ and simplify.

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Step 3.2.1

Divide each term in $5 y = 6$ by $5$ .

$\frac{5 y}{5} = \frac{6}{5}$

$x = - 1 + 2 y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{6}{5}$

$x = - 1 + 2 y$

Step 3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{6}{5}$

$x = - 1 + 2 y$

$y = \frac{6}{5}$

$x = - 1 + 2 y$

$y = \frac{6}{5}$

$x = - 1 + 2 y$

$y = \frac{6}{5}$

$x = - 1 + 2 y$

$y = \frac{6}{5}$

$x = - 1 + 2 y$

Step 4

Replace all occurrences of $y$ with $\frac{6}{5}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - 1 + 2 y$ with $\frac{6}{5}$ .

$x = - 1 + 2 (\frac{6}{5})$

$y = \frac{6}{5}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 1 + 2 (\frac{6}{5})$ .

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Step 4.2.1.1

Multiply $2 (\frac{6}{5})$ .

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Step 4.2.1.1.1

Combine $2$ and $\frac{6}{5}$ .

$x = - 1 + \frac{2 \cdot 6}{5}$

$y = \frac{6}{5}$

Step 4.2.1.1.2

Multiply $2$ by $6$ .

$x = - 1 + \frac{12}{5}$

$y = \frac{6}{5}$

$x = - 1 + \frac{12}{5}$

$y = \frac{6}{5}$

Step 4.2.1.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$x = - 1 \cdot \frac{5}{5} + \frac{12}{5}$

$y = \frac{6}{5}$

Step 4.2.1.3

Combine $- 1$ and $\frac{5}{5}$ .

$x = \frac{- 1 \cdot 5}{5} + \frac{12}{5}$

$y = \frac{6}{5}$

Step 4.2.1.4

Combine the numerators over the common denominator.

$x = \frac{- 1 \cdot 5 + 12}{5}$

$y = \frac{6}{5}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply $- 1$ by $5$ .

$x = \frac{- 5 + 12}{5}$

$y = \frac{6}{5}$

Step 4.2.1.5.2

Add $- 5$ and $12$ .

$x = \frac{7}{5}$

$y = \frac{6}{5}$

$x = \frac{7}{5}$

$y = \frac{6}{5}$

$x = \frac{7}{5}$

$y = \frac{6}{5}$

$x = \frac{7}{5}$

$y = \frac{6}{5}$

$x = \frac{7}{5}$

$y = \frac{6}{5}$

Step 5

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

$(\frac{7}{5}, \frac{6}{5})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{7}{5}, \frac{6}{5})$

Equation Form:

$x = \frac{7}{5}, y = \frac{6}{5}$

Step 7