Linear Algebra Examples

x + 2 y = 4

$x + 2 y = 4$ ,

x - y = - 3

$x - y = - 3$

Step 1

Subtract

2 y

$2 y$ from both sides of the equation.

x = 4 - 2 y

$x = 4 - 2 y$

x - y = - 3

$x - y = - 3$

Step 2

Replace all occurrences of

x

$x$ with

4 - 2 y

$4 - 2 y$ in each equation.

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Step 2.1

Replace all occurrences of

x

$x$ in

x - y = - 3

$x - y = - 3$ with

4 - 2 y

$4 - 2 y$ .

(4 - 2 y) - y = - 3

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

x = 4 - 2 y

$x = 4 - 2 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Subtract

y

$y$ from

- 2 y

$- 2 y$ .

4 - 3 y = - 3

$4 - 3 y = - 3$

x = 4 - 2 y

$x = 4 - 2 y$

4 - 3 y = - 3

$4 - 3 y = - 3$

x = 4 - 2 y

$x = 4 - 2 y$

4 - 3 y = - 3

$4 - 3 y = - 3$

x = 4 - 2 y

$x = 4 - 2 y$

Step 3

Solve for

y

$y$ in

4 - 3 y = - 3

$4 - 3 y = - 3$ .

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Step 3.1

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 3.1.1

Subtract

4

$4$ from both sides of the equation.

- 3 y = - 3 - 4

$- 3 y = - 3 - 4$

x = 4 - 2 y

$x = 4 - 2 y$

Step 3.1.2

Subtract

4

$4$ from

- 3

$- 3$ .

- 3 y = - 7

$- 3 y = - 7$

x = 4 - 2 y

$x = 4 - 2 y$

- 3 y = - 7

$- 3 y = - 7$

x = 4 - 2 y

$x = 4 - 2 y$

Step 3.2

Divide each term in

- 3 y = - 7

$- 3 y = - 7$ by

- 3

$- 3$ and simplify.

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Step 3.2.1

Divide each term in

- 3 y = - 7

$- 3 y = - 7$ by

- 3

$- 3$ .

\frac{- 3 y}{- 3} = \frac{- 7}{- 3}

$\frac{- 3 y}{- 3} = \frac{- 7}{- 3}$

x = 4 - 2 y

$x = 4 - 2 y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of

- 3

$- 3$ .

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Step 3.2.2.1.1

Cancel the common factor.

\frac{- 3 y}{- 3} = \frac{- 7}{- 3}

$\frac{- 3 y}{- 3} = \frac{- 7}{- 3}$

x = 4 - 2 y

$x = 4 - 2 y$

Step 3.2.2.1.2

Divide

y

$y$ by

1

$1$ .

y = \frac{- 7}{- 3}

$y = \frac{- 7}{- 3}$

x = 4 - 2 y

$x = 4 - 2 y$

y = \frac{- 7}{- 3}

$y = \frac{- 7}{- 3}$

x = 4 - 2 y

$x = 4 - 2 y$

y = \frac{- 7}{- 3}

$y = \frac{- 7}{- 3}$

x = 4 - 2 y

$x = 4 - 2 y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Dividing two negative values results in a positive value.

y = \frac{7}{3}

$y = \frac{7}{3}$

x = 4 - 2 y

$x = 4 - 2 y$

y = \frac{7}{3}

$y = \frac{7}{3}$

x = 4 - 2 y

$x = 4 - 2 y$

y = \frac{7}{3}

$y = \frac{7}{3}$

x = 4 - 2 y

$x = 4 - 2 y$

y = \frac{7}{3}

$y = \frac{7}{3}$

x = 4 - 2 y

$x = 4 - 2 y$

Step 4

Replace all occurrences of

y

$y$ with

\frac{7}{3}

$\frac{7}{3}$ in each equation.

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Step 4.1

Replace all occurrences of

y

$y$ in

x = 4 - 2 y

$x = 4 - 2 y$ with

\frac{7}{3}

$\frac{7}{3}$ .

x = 4 - 2 (\frac{7}{3})

$x = 4 - 2 (\frac{7}{3})$

y = \frac{7}{3}

$y = \frac{7}{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify

4 - 2 (\frac{7}{3})

$4 - 2 (\frac{7}{3})$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Multiply

- 2 (\frac{7}{3})

$- 2 (\frac{7}{3})$ .

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Step 4.2.1.1.1.1

Combine

- 2

$- 2$ and

\frac{7}{3}

$\frac{7}{3}$ .

x = 4 + \frac{- 2 \cdot 7}{3}

$x = 4 + \frac{- 2 \cdot 7}{3}$

y = \frac{7}{3}

$y = \frac{7}{3}$

Step 4.2.1.1.1.2

Multiply

- 2

$- 2$ by

7

$7$ .

x = 4 + \frac{- 14}{3}

$x = 4 + \frac{- 14}{3}$

y = \frac{7}{3}

$y = \frac{7}{3}$

x = 4 + \frac{- 14}{3}

$x = 4 + \frac{- 14}{3}$

y = \frac{7}{3}

$y = \frac{7}{3}$

Step 4.2.1.1.2

Move the negative in front of the fraction.

x = 4 - \frac{14}{3}

$x = 4 - \frac{14}{3}$

y = \frac{7}{3}

$y = \frac{7}{3}$

x = 4 - \frac{14}{3}

$x = 4 - \frac{14}{3}$

y = \frac{7}{3}

$y = \frac{7}{3}$

Step 4.2.1.2

To write

4

$4$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

x = 4 \cdot \frac{3}{3} - \frac{14}{3}

$x = 4 \cdot \frac{3}{3} - \frac{14}{3}$

y = \frac{7}{3}

$y = \frac{7}{3}$

Step 4.2.1.3

Combine

4

$4$ and

\frac{3}{3}

$\frac{3}{3}$ .

x = \frac{4 \cdot 3}{3} - \frac{14}{3}

$x = \frac{4 \cdot 3}{3} - \frac{14}{3}$

y = \frac{7}{3}

$y = \frac{7}{3}$

Step 4.2.1.4

Combine the numerators over the common denominator.

x = \frac{4 \cdot 3 - 14}{3}

$x = \frac{4 \cdot 3 - 14}{3}$

y = \frac{7}{3}

$y = \frac{7}{3}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply

4

$4$ by

3

$3$ .

x = \frac{12 - 14}{3}

$x = \frac{12 - 14}{3}$

y = \frac{7}{3}

$y = \frac{7}{3}$

Step 4.2.1.5.2

Subtract

14

$14$ from

12

$12$ .

x = \frac{- 2}{3}

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

y = \frac{7}{3}

$y = \frac{7}{3}$

x = \frac{- 2}{3}

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

y = \frac{7}{3}

$y = \frac{7}{3}$

Step 4.2.1.6

Move the negative in front of the fraction.

x = - \frac{2}{3}

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

y = \frac{7}{3}

$y = \frac{7}{3}$

x = - \frac{2}{3}

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

y = \frac{7}{3}

$y = \frac{7}{3}$

x = - \frac{2}{3}

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

y = \frac{7}{3}

$y = \frac{7}{3}$

x = - \frac{2}{3}

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

y = \frac{7}{3}

$y = \frac{7}{3}$

Step 5

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

(- \frac{2}{3}, \frac{7}{3})

$(- \frac{2}{3}, \frac{7}{3})$

Step 6

The result can be shown in multiple forms.

Point Form:

(- \frac{2}{3}, \frac{7}{3})

$(- \frac{2}{3}, \frac{7}{3})$

Equation Form:

x = - \frac{2}{3}, y = \frac{7}{3}

$x = - \frac{2}{3}, y = \frac{7}{3}$

Step 7

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