Algebra Examples

$x y = 6$ $3 y = x - 3$

Step 1

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

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Step 1.1

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

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

$3 y = x - 3$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $y$ .

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Step 1.2.1.1

Cancel the common factor.

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

$3 y = x - 3$

Step 1.2.1.2

Divide $x$ by $1$ .

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

$3 y = x - 3$

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

$3 y = x - 3$

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

$3 y = x - 3$

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

$3 y = x - 3$

Step 2

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

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Step 2.1

Replace all occurrences of $x$ in $3 y = x - 3$ with $\frac{6}{y}$ .

$3 y = (\frac{6}{y}) - 3$

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

Step 2.2

Simplify the right side.

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Step 2.2.1

Remove parentheses.

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

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

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

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

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

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

Step 3

Solve for $y$ in $3 y = \frac{6}{y} - 3$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$1, y, 1$

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

Step 3.1.2

The LCM of one and any expression is the expression.

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

Step 3.2

Multiply each term in $3 y = \frac{6}{y} - 3$ by $y$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $3 y = \frac{6}{y} - 3$ by $y$ .

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

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

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Multiply $y$ by $y$ by adding the exponents.

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Step 3.2.2.1.1

Move $y$ .

$3 (y \cdot y) = \frac{6}{y} \cdot y - 3 y$

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

Step 3.2.2.1.2

Multiply $y$ by $y$ .

$3 y^{2} = \frac{6}{y} \cdot y - 3 y$

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

$3 y^{2} = \frac{6}{y} \cdot y - 3 y$

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

$3 y^{2} = \frac{6}{y} \cdot y - 3 y$

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

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Cancel the common factor of $y$ .

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Step 3.2.3.1.1

Cancel the common factor.

$3 y^{2} = \frac{6}{y} \cdot y - 3 y$

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

Step 3.2.3.1.2

Rewrite the expression.

$3 y^{2} = 6 - 3 y$

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

$3 y^{2} = 6 - 3 y$

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

$3 y^{2} = 6 - 3 y$

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

$3 y^{2} = 6 - 3 y$

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

Step 3.3

Solve the equation.

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Step 3.3.1

Add $3 y$ to both sides of the equation.

$3 y^{2} + 3 y = 6$

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

Step 3.3.2

Subtract $6$ from both sides of the equation.

$3 y^{2} + 3 y - 6 = 0$

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

Step 3.3.3

Factor the left side of the equation.

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Step 3.3.3.1

Factor $3$ out of $3 y^{2} + 3 y - 6$ .

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Step 3.3.3.1.1

Factor $3$ out of $3 y^{2}$ .

$3 (y^{2}) + 3 y - 6 = 0$

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

Step 3.3.3.1.2

Factor $3$ out of $3 y$ .

$3 (y^{2}) + 3 (y) - 6 = 0$

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

Step 3.3.3.1.3

Factor $3$ out of $- 6$ .

$3 (y^{2}) + 3 y + 3 \cdot - 2 = 0$

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

Step 3.3.3.1.4

Factor $3$ out of $3 (y^{2}) + 3 y$ .

$3 (y^{2} + y) + 3 \cdot - 2 = 0$

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

Step 3.3.3.1.5

Factor $3$ out of $3 (y^{2} + y) + 3 \cdot - 2$ .

$3 (y^{2} + y - 2) = 0$

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

$3 (y^{2} + y - 2) = 0$

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

Step 3.3.3.2

Factor.

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Step 3.3.3.2.1

Factor $y^{2} + y - 2$ using the AC method.

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Step 3.3.3.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

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

Step 3.3.3.2.1.2

Write the factored form using these integers.

$3 ((y - 1) (y + 2)) = 0$

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

$3 ((y - 1) (y + 2)) = 0$

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

Step 3.3.3.2.2

Remove unnecessary parentheses.

$3 (y - 1) (y + 2) = 0$

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

$3 (y - 1) (y + 2) = 0$

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

$3 (y - 1) (y + 2) = 0$

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

Step 3.3.4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$y - 1 = 0$

$y + 2 = 0$

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

Step 3.3.5

Set $y - 1$ equal to $0$ and solve for $y$ .

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Step 3.3.5.1

Set $y - 1$ equal to $0$ .

$y - 1 = 0$

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

Step 3.3.5.2

Add $1$ to both sides of the equation.

$y = 1$

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

$y = 1$

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

Step 3.3.6

Set $y + 2$ equal to $0$ and solve for $y$ .

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Step 3.3.6.1

Set $y + 2$ equal to $0$ .

$y + 2 = 0$

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

Step 3.3.6.2

Subtract $2$ from both sides of the equation.

$y = - 2$

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

$y = - 2$

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

Step 3.3.7

The final solution is all the values that make $3 (y - 1) (y + 2) = 0$ true.

$y = 1, - 2$

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

$y = 1, - 2$

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

$y = 1, - 2$

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

Step 4

Replace all occurrences of $y$ with $1$ in each equation.

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Step 4.1

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

$x = \frac{6}{1}$

$y = 1$

Step 4.2

Simplify the right side.

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Step 4.2.1

Divide $6$ by $1$ .

$x = 6$

$y = 1$

$x = 6$

$y = 1$

$x = 6$

$y = 1$

Step 5

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

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Step 5.1

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

$x = \frac{6}{- 2}$

$y = - 2$

Step 5.2

Simplify the right side.

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Step 5.2.1

Divide $6$ by $- 2$ .

$x = - 3$

$y = - 2$

$x = - 3$

$y = - 2$

$x = - 3$

$y = - 2$

Step 6

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

$(6, 1)$

$(- 3, - 2)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(6, 1), (- 3, - 2)$

Equation Form:

$x = 6, y = 1$

$x = - 3, y = - 2$

Step 8