Examples

x + y = 3

$x + y = 3$ ,

2 x - 8 y = 19

$2 x - 8 y = 19$

Step 1

Subtract

y

$y$ from both sides of the equation.

x = 3 - y

$x = 3 - y$

2 x - 8 y = 19

$2 x - 8 y = 19$

Step 2

Replace all occurrences of

x

$x$ with

3 - y

$3 - y$ in each equation.

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Step 2.1

Replace all occurrences of

x

$x$ in

2 x - 8 y = 19

$2 x - 8 y = 19$ with

3 - y

$3 - y$ .

2 (3 - y) - 8 y = 19

$2 (3 - y) - 8 y = 19$

x = 3 - y

$x = 3 - y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify

2 (3 - y) - 8 y

$2 (3 - y) - 8 y$ .

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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 3 + 2 (- y) - 8 y = 19

$2 \cdot 3 + 2 (- y) - 8 y = 19$

x = 3 - y

$x = 3 - y$

Step 2.2.1.1.2

Multiply

2

$2$ by

3

$3$ .

6 + 2 (- y) - 8 y = 19

$6 + 2 (- y) - 8 y = 19$

x = 3 - y

$x = 3 - y$

Step 2.2.1.1.3

Multiply

- 1

$- 1$ by

2

$2$ .

6 - 2 y - 8 y = 19

$6 - 2 y - 8 y = 19$

x = 3 - y

$x = 3 - y$

6 - 2 y - 8 y = 19

$6 - 2 y - 8 y = 19$

x = 3 - y

$x = 3 - y$

Step 2.2.1.2

Subtract

8 y

$8 y$ from

- 2 y

$- 2 y$ .

6 - 10 y = 19

$6 - 10 y = 19$

x = 3 - y

$x = 3 - y$

6 - 10 y = 19

$6 - 10 y = 19$

x = 3 - y

$x = 3 - y$

6 - 10 y = 19

$6 - 10 y = 19$

x = 3 - y

$x = 3 - y$

6 - 10 y = 19

$6 - 10 y = 19$

x = 3 - y

$x = 3 - y$

Step 3

Solve for

y

$y$ in

6 - 10 y = 19

$6 - 10 y = 19$ .

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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

6

$6$ from both sides of the equation.

- 10 y = 19 - 6

$- 10 y = 19 - 6$

x = 3 - y

$x = 3 - y$

Step 3.1.2

Subtract

6

$6$ from

19

$19$ .

- 10 y = 13

$- 10 y = 13$

x = 3 - y

$x = 3 - y$

- 10 y = 13

$- 10 y = 13$

x = 3 - y

$x = 3 - y$

Step 3.2

Divide each term in

- 10 y = 13

$- 10 y = 13$ by

- 10

$- 10$ and simplify.

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Step 3.2.1

Divide each term in

- 10 y = 13

$- 10 y = 13$ by

- 10

$- 10$ .

\frac{- 10 y}{- 10} = \frac{13}{- 10}

$\frac{- 10 y}{- 10} = \frac{13}{- 10}$

x = 3 - y

$x = 3 - y$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of

- 10

$- 10$ .

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Step 3.2.2.1.1

Cancel the common factor.

\frac{- 10 y}{- 10} = \frac{13}{- 10}

$\frac{- 10 y}{- 10} = \frac{13}{- 10}$

x = 3 - y

$x = 3 - y$

Step 3.2.2.1.2

Divide

y

$y$ by

1

$1$ .

y = \frac{13}{- 10}

$y = \frac{13}{- 10}$

x = 3 - y

$x = 3 - y$

y = \frac{13}{- 10}

$y = \frac{13}{- 10}$

x = 3 - y

$x = 3 - y$

y = \frac{13}{- 10}

$y = \frac{13}{- 10}$

x = 3 - y

$x = 3 - y$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Move the negative in front of the fraction.

y = - \frac{13}{10}

$y = - \frac{13}{10}$

x = 3 - y

$x = 3 - y$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

x = 3 - y

$x = 3 - y$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

x = 3 - y

$x = 3 - y$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

x = 3 - y

$x = 3 - y$

Step 4

Replace all occurrences of

y

$y$ with

- \frac{13}{10}

$- \frac{13}{10}$ in each equation.

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Step 4.1

Replace all occurrences of

y

$y$ in

x = 3 - y

$x = 3 - y$ with

- \frac{13}{10}

$- \frac{13}{10}$ .

x = 3 - (- \frac{13}{10})

$x = 3 - (- \frac{13}{10})$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify

3 - (- \frac{13}{10})

$3 - (- \frac{13}{10})$ .

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Step 4.2.1.1

Multiply

- (- \frac{13}{10})

$- (- \frac{13}{10})$ .

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Step 4.2.1.1.1

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

x = 3 + 1 (\frac{13}{10})

$x = 3 + 1 (\frac{13}{10})$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

Step 4.2.1.1.2

Multiply

\frac{13}{10}

$\frac{13}{10}$ by

1

$1$ .

x = 3 + \frac{13}{10}

$x = 3 + \frac{13}{10}$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

x = 3 + \frac{13}{10}

$x = 3 + \frac{13}{10}$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

Step 4.2.1.2

To write

3

$3$ as a fraction with a common denominator, multiply by

\frac{10}{10}

$\frac{10}{10}$ .

x = 3 \cdot \frac{10}{10} + \frac{13}{10}

$x = 3 \cdot \frac{10}{10} + \frac{13}{10}$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

Step 4.2.1.3

Combine

3

$3$ and

\frac{10}{10}

$\frac{10}{10}$ .

x = \frac{3 \cdot 10}{10} + \frac{13}{10}

$x = \frac{3 \cdot 10}{10} + \frac{13}{10}$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

Step 4.2.1.4

Combine the numerators over the common denominator.

x = \frac{3 \cdot 10 + 13}{10}

$x = \frac{3 \cdot 10 + 13}{10}$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply

3

$3$ by

10

$10$ .

x = \frac{30 + 13}{10}

$x = \frac{30 + 13}{10}$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

Step 4.2.1.5.2

Add

30

$30$ and

13

$13$ .

x = \frac{43}{10}

$x = \frac{43}{10}$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

x = \frac{43}{10}

$x = \frac{43}{10}$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

x = \frac{43}{10}

$x = \frac{43}{10}$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

x = \frac{43}{10}

$x = \frac{43}{10}$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

x = \frac{43}{10}

$x = \frac{43}{10}$

y = - \frac{13}{10}

$y = - \frac{13}{10}$

Step 5

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

(\frac{43}{10}, - \frac{13}{10})

$(\frac{43}{10}, - \frac{13}{10})$

Step 6

The result can be shown in multiple forms.

Point Form:

(\frac{43}{10}, - \frac{13}{10})

$(\frac{43}{10}, - \frac{13}{10})$

Equation Form:

x = \frac{43}{10}, y = - \frac{13}{10}

$x = \frac{43}{10}, y = - \frac{13}{10}$

Step 7

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