Algebra Examples

- 2 x - 3 y + 5 z = 7

$- 2 x - 3 y + 5 z = 7$

Step 1

Solve the equation for

y

$y$ .

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Step 1.1

Move all terms not containing

y

$y$ to the right side of the equation.

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Step 1.1.1

Add

2 x

$2 x$ to both sides of the equation.

- 3 y + 5 z = 7 + 2 x

$- 3 y + 5 z = 7 + 2 x$

Step 1.1.2

Subtract

5 z

$5 z$ from both sides of the equation.

- 3 y = 7 + 2 x - 5 z

$- 3 y = 7 + 2 x - 5 z$

- 3 y = 7 + 2 x - 5 z

$- 3 y = 7 + 2 x - 5 z$

Step 1.2

Divide each term in

- 3 y = 7 + 2 x - 5 z

$- 3 y = 7 + 2 x - 5 z$ by

- 3

$- 3$ and simplify.

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Step 1.2.1

Divide each term in

- 3 y = 7 + 2 x - 5 z

$- 3 y = 7 + 2 x - 5 z$ by

- 3

$- 3$ .

\frac{- 3 y}{- 3} = \frac{7}{- 3} + \frac{2 x}{- 3} + \frac{- 5 z}{- 3}

$\frac{- 3 y}{- 3} = \frac{7}{- 3} + \frac{2 x}{- 3} + \frac{- 5 z}{- 3}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of

- 3

$- 3$ .

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Step 1.2.2.1.1

Cancel the common factor.

\frac{- 3 y}{- 3} = \frac{7}{- 3} + \frac{2 x}{- 3} + \frac{- 5 z}{- 3}

$\frac{- 3 y}{- 3} = \frac{7}{- 3} + \frac{2 x}{- 3} + \frac{- 5 z}{- 3}$

Step 1.2.2.1.2

Divide

y

$y$ by

1

$1$ .

y = \frac{7}{- 3} + \frac{2 x}{- 3} + \frac{- 5 z}{- 3}

$y = \frac{7}{- 3} + \frac{2 x}{- 3} + \frac{- 5 z}{- 3}$

y = \frac{7}{- 3} + \frac{2 x}{- 3} + \frac{- 5 z}{- 3}

$y = \frac{7}{- 3} + \frac{2 x}{- 3} + \frac{- 5 z}{- 3}$

y = \frac{7}{- 3} + \frac{2 x}{- 3} + \frac{- 5 z}{- 3}

$y = \frac{7}{- 3} + \frac{2 x}{- 3} + \frac{- 5 z}{- 3}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Move the negative in front of the fraction.

y = - \frac{7}{3} + \frac{2 x}{- 3} + \frac{- 5 z}{- 3}

$y = - \frac{7}{3} + \frac{2 x}{- 3} + \frac{- 5 z}{- 3}$

Step 1.2.3.1.2

Move the negative in front of the fraction.

y = - \frac{7}{3} - \frac{2 x}{3} + \frac{- 5 z}{- 3}

$y = - \frac{7}{3} - \frac{2 x}{3} + \frac{- 5 z}{- 3}$

Step 1.2.3.1.3

Dividing two negative values results in a positive value.

y = - \frac{7}{3} - \frac{2 x}{3} + \frac{5 z}{3}

$y = - \frac{7}{3} - \frac{2 x}{3} + \frac{5 z}{3}$

y = - \frac{7}{3} - \frac{2 x}{3} + \frac{5 z}{3}

$y = - \frac{7}{3} - \frac{2 x}{3} + \frac{5 z}{3}$

y = - \frac{7}{3} - \frac{2 x}{3} + \frac{5 z}{3}

$y = - \frac{7}{3} - \frac{2 x}{3} + \frac{5 z}{3}$

y = - \frac{7}{3} - \frac{2 x}{3} + \frac{5 z}{3}

$y = - \frac{7}{3} - \frac{2 x}{3} + \frac{5 z}{3}$

y = - \frac{7}{3} - \frac{2 x}{3} + \frac{5 z}{3}

$y = - \frac{7}{3} - \frac{2 x}{3} + \frac{5 z}{3}$

Step 2

Choose any values for

x

$x$ and

y

$y$ that are in the domain to plug into the equation.

Step 3

Choose

0

$0$ to substitute in for

x

$x$ and

1

$1$ to substitute for

z

$z$ .

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Step 3.1

Remove parentheses.

y = - \frac{7}{3} - \frac{2 (0)}{3} + \frac{5 (1)}{3}

$y = - \frac{7}{3} - \frac{2 (0)}{3} + \frac{5 (1)}{3}$

Step 3.2

Simplify

- \frac{7}{3} - \frac{2 (0)}{3} + \frac{5 (1)}{3}

$- \frac{7}{3} - \frac{2 (0)}{3} + \frac{5 (1)}{3}$ .

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Step 3.2.1

Combine the numerators over the common denominator.

y = \frac{- 7 - 2 \cdot 0 + 5 (1)}{3}

$y = \frac{- 7 - 2 \cdot 0 + 5 (1)}{3}$

Step 3.2.2

Simplify each term.

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Step 3.2.2.1

Multiply

- 2

$- 2$ by

0

$0$ .

y = \frac{- 7 + 0 + 5 (1)}{3}

$y = \frac{- 7 + 0 + 5 (1)}{3}$

Step 3.2.2.2

Multiply

5

$5$ by

1

$1$ .

y = \frac{- 7 + 0 + 5}{3}

$y = \frac{- 7 + 0 + 5}{3}$

y = \frac{- 7 + 0 + 5}{3}

$y = \frac{- 7 + 0 + 5}{3}$

Step 3.2.3

Simplify the expression.

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Step 3.2.3.1

Add

- 7

$- 7$ and

0

$0$ .

y = \frac{- 7 + 5}{3}

$y = \frac{- 7 + 5}{3}$

Step 3.2.3.2

Add

- 7

$- 7$ and

5

$5$ .

y = \frac{- 2}{3}

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

Step 3.2.3.3

Move the negative in front of the fraction.

y = - \frac{2}{3}

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

y = - \frac{2}{3}

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

y = - \frac{2}{3}

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

Step 3.3

Use the

x

$x$ ,

y

$y$ , and

z

$z$ values to form the ordered pair.

(0, - \frac{2}{3}, 1)

$(0, - \frac{2}{3}, 1)$

(0, - \frac{2}{3}, 1)

$(0, - \frac{2}{3}, 1)$

Step 4

Choose

1

$1$ to substitute in for

x

$x$ and

2

$2$ to substitute for

z

$z$ .

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Step 4.1

Remove parentheses.

y = - \frac{7}{3} - \frac{2 (1)}{3} + \frac{5 (2)}{3}

$y = - \frac{7}{3} - \frac{2 (1)}{3} + \frac{5 (2)}{3}$

Step 4.2

Simplify

- \frac{7}{3} - \frac{2 (1)}{3} + \frac{5 (2)}{3}

$- \frac{7}{3} - \frac{2 (1)}{3} + \frac{5 (2)}{3}$ .

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Step 4.2.1

Combine the numerators over the common denominator.

y = \frac{- 7 - 2 \cdot 1 + 5 (2)}{3}

$y = \frac{- 7 - 2 \cdot 1 + 5 (2)}{3}$

Step 4.2.2

Simplify each term.

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Step 4.2.2.1

Multiply

- 2

$- 2$ by

1

$1$ .

y = \frac{- 7 - 2 + 5 (2)}{3}

$y = \frac{- 7 - 2 + 5 (2)}{3}$

Step 4.2.2.2

Multiply

5

$5$ by

2

$2$ .

y = \frac{- 7 - 2 + 10}{3}

$y = \frac{- 7 - 2 + 10}{3}$

y = \frac{- 7 - 2 + 10}{3}

$y = \frac{- 7 - 2 + 10}{3}$

Step 4.2.3

Simplify by adding and subtracting.

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Step 4.2.3.1

Subtract

2

$2$ from

- 7

$- 7$ .

y = \frac{- 9 + 10}{3}

$y = \frac{- 9 + 10}{3}$

Step 4.2.3.2

Add

- 9

$- 9$ and

10

$10$ .

y = \frac{1}{3}

$y = \frac{1}{3}$

y = \frac{1}{3}

$y = \frac{1}{3}$

y = \frac{1}{3}

$y = \frac{1}{3}$

Step 4.3

Use the

x

$x$ ,

y

$y$ , and

z

$z$ values to form the ordered pair.

(1, \frac{1}{3}, 2)

$(1, \frac{1}{3}, 2)$

(1, \frac{1}{3}, 2)

$(1, \frac{1}{3}, 2)$

Step 5

Choose

2

$2$ to substitute in for

x

$x$ and

3

$3$ to substitute for

z

$z$ .

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Step 5.1

Remove parentheses.

y = - \frac{7}{3} - \frac{2 (2)}{3} + \frac{5 (3)}{3}

$y = - \frac{7}{3} - \frac{2 (2)}{3} + \frac{5 (3)}{3}$

Step 5.2

Simplify

- \frac{7}{3} - \frac{2 (2)}{3} + \frac{5 (3)}{3}

$- \frac{7}{3} - \frac{2 (2)}{3} + \frac{5 (3)}{3}$ .

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Step 5.2.1

Combine the numerators over the common denominator.

y = \frac{- 7 - 2 \cdot 2 + 5 (3)}{3}

$y = \frac{- 7 - 2 \cdot 2 + 5 (3)}{3}$

Step 5.2.2

Simplify each term.

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Step 5.2.2.1

Multiply

- 2

$- 2$ by

2

$2$ .

y = \frac{- 7 - 4 + 5 (3)}{3}

$y = \frac{- 7 - 4 + 5 (3)}{3}$

Step 5.2.2.2

Multiply

5

$5$ by

3

$3$ .

y = \frac{- 7 - 4 + 15}{3}

$y = \frac{- 7 - 4 + 15}{3}$

y = \frac{- 7 - 4 + 15}{3}

$y = \frac{- 7 - 4 + 15}{3}$

Step 5.2.3

Simplify by adding and subtracting.

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Step 5.2.3.1

Subtract

4

$4$ from

- 7

$- 7$ .

y = \frac{- 11 + 15}{3}

$y = \frac{- 11 + 15}{3}$

Step 5.2.3.2

Add

- 11

$- 11$ and

15

$15$ .

y = \frac{4}{3}

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

y = \frac{4}{3}

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

y = \frac{4}{3}

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

Step 5.3

Use the

x

$x$ ,

y

$y$ , and

z

$z$ values to form the ordered pair.

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

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

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

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

Step 6

These are three possible solutions to the equation.

(0, - \frac{2}{3}, 1), (1, \frac{1}{3}, 2), (2, \frac{4}{3}, 3)

$(0, - \frac{2}{3}, 1), (1, \frac{1}{3}, 2), (2, \frac{4}{3}, 3)$

⎡ ⎢ ⎣ \begin{matrix} x^{2} & \frac{1}{2} \sqrt{π} & \int x d x \end{matrix} ⎤ ⎥ ⎦

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$