Linear Algebra Examples

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

Step 1

Subtract $x$ from both sides of the equation.

$y = 2 - x$

$4 = x + 4 y$

Step 2

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

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Step 2.1

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

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

$y = 2 - x$

Step 2.2

Simplify the right side.

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Step 2.2.1

Simplify $x + 4 (2 - x)$ .

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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.

$4 = x + 4 \cdot 2 + 4 (- x)$

$y = 2 - x$

Step 2.2.1.1.2

Multiply $4$ by $2$ .

$4 = x + 8 + 4 (- x)$

$y = 2 - x$

Step 2.2.1.1.3

Multiply $- 1$ by $4$ .

$4 = x + 8 - 4 x$

$y = 2 - x$

$4 = x + 8 - 4 x$

$y = 2 - x$

Step 2.2.1.2

Subtract $4 x$ from $x$ .

$4 = - 3 x + 8$

$y = 2 - x$

$4 = - 3 x + 8$

$y = 2 - x$

$4 = - 3 x + 8$

$y = 2 - x$

$4 = - 3 x + 8$

$y = 2 - x$

Step 3

Solve for $x$ in $4 = - 3 x + 8$ .

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Step 3.1

Rewrite the equation as $- 3 x + 8 = 4$ .

$- 3 x + 8 = 4$

$y = 2 - x$

Step 3.2

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

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Step 3.2.1

Subtract $8$ from both sides of the equation.

$- 3 x = 4 - 8$

$y = 2 - x$

Step 3.2.2

Subtract $8$ from $4$ .

$- 3 x = - 4$

$y = 2 - x$

$- 3 x = - 4$

$y = 2 - x$

Step 3.3

Divide each term in $- 3 x = - 4$ by $- 3$ and simplify.

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Step 3.3.1

Divide each term in $- 3 x = - 4$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- 4}{- 3}$

$y = 2 - x$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $- 3$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- 4}{- 3}$

$y = 2 - x$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{- 3}$

$y = 2 - x$

$x = \frac{- 4}{- 3}$

$y = 2 - x$

$x = \frac{- 4}{- 3}$

$y = 2 - x$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Dividing two negative values results in a positive value.

$x = \frac{4}{3}$

$y = 2 - x$

$x = \frac{4}{3}$

$y = 2 - x$

$x = \frac{4}{3}$

$y = 2 - x$

$x = \frac{4}{3}$

$y = 2 - x$

Step 4

Replace all occurrences of $x$ with $\frac{4}{3}$ in each equation.

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Step 4.1

Replace all occurrences of $x$ in $y = 2 - x$ with $\frac{4}{3}$ .

$y = 2 - (\frac{4}{3})$

$x = \frac{4}{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $2 - (\frac{4}{3})$ .

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Step 4.2.1.1

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$y = 2 \cdot \frac{3}{3} - \frac{4}{3}$

$x = \frac{4}{3}$

Step 4.2.1.2

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

$y = \frac{2 \cdot 3}{3} - \frac{4}{3}$

$x = \frac{4}{3}$

Step 4.2.1.3

Combine the numerators over the common denominator.

$y = \frac{2 \cdot 3 - 4}{3}$

$x = \frac{4}{3}$

Step 4.2.1.4

Simplify the numerator.

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Step 4.2.1.4.1

Multiply $2$ by $3$ .

$y = \frac{6 - 4}{3}$

$x = \frac{4}{3}$

Step 4.2.1.4.2

Subtract $4$ from $6$ .

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

$x = \frac{4}{3}$

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

$x = \frac{4}{3}$

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

$x = \frac{4}{3}$

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

$x = \frac{4}{3}$

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

$x = \frac{4}{3}$

Step 5

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

$(\frac{4}{3}, \frac{2}{3})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(\frac{4}{3}, \frac{2}{3})$

Equation Form:

$x = \frac{4}{3}, y = \frac{2}{3}$

Step 7