Precalculus Examples

$\frac{2}{x} - \frac{3}{y} = 1$ , $\frac{4}{x} + \frac{7}{y} = 1$

Step 1

Multiply each term in $\frac{2}{x} - \frac{3}{y} = 1$ by $xy$ to eliminate the fractions.

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Step 1.1

Multiply each term in $\frac{2}{x} - \frac{3}{y} = 1$ by $xy$ .

$\frac{2}{x} \cdot (x y) - \frac{3}{y} \cdot (x y) = 1 x y$

$\frac{4}{x} + \frac{7}{y} = 1$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Combine $\frac{2}{x}$ and $xy$ .

$\frac{2 x y}{x} - \frac{3}{y} \cdot (x y) = 1 x y$

$\frac{4}{x} + \frac{7}{y} = 1$

Step 1.2.1.2

Combine $xy$ and $\frac{3}{y}$ .

$\frac{2 x y}{x} - \frac{x y \cdot 3}{y} = 1 x y$

$\frac{4}{x} + \frac{7}{y} = 1$

Step 1.2.1.3

Move $3$ to the left of $xy$ .

$\frac{2 x y}{x} - \frac{3 x y}{y} = 1 x y$

$\frac{4}{x} + \frac{7}{y} = 1$

$\frac{2 x y}{x} - \frac{3 x y}{y} = 1 x y$

$\frac{4}{x} + \frac{7}{y} = 1$

$\frac{2 x y}{x} - \frac{3 x y}{y} = 1 x y$

$\frac{4}{x} + \frac{7}{y} = 1$

Step 1.3

Simplify the right side.

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Step 1.3.1

Multiply $xy$ by $1$ .

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

$\frac{4}{x} + \frac{7}{y} = 1$

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

$\frac{4}{x} + \frac{7}{y} = 1$

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

$\frac{4}{x} + \frac{7}{y} = 1$

Step 2

Multiply each term in $\frac{4}{x} + \frac{7}{y} = 1$ by $xy$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{4}{x} + \frac{7}{y} = 1$ by $xy$ .

$\frac{4}{x} \cdot (x y) + \frac{7}{y} \cdot (x y) = 1 x y$

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

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Combine $\frac{4}{x}$ and $xy$ .

$\frac{4 x y}{x} + \frac{7}{y} \cdot (x y) = 1 x y$

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

Step 2.2.1.2

Combine $\frac{7}{y}$ and $xy$ .

$\frac{4 x y}{x} + \frac{7 x y}{y} = 1 x y$

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

$\frac{4 x y}{x} + \frac{7 x y}{y} = 1 x y$

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

$\frac{4 x y}{x} + \frac{7 x y}{y} = 1 x y$

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

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $xy$ by $1$ .

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

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

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

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

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

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

Step 3

Simplify the left side.

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Step 3.1

Simplify $\frac{2 x y}{x} - \frac{3 x y}{y}$ .

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Cancel the common factor of $x$ .

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Step 3.1.1.1.1

Cancel the common factor.

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

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

Step 3.1.1.1.2

Divide $2 y$ by $1$ .

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

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

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

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

Step 3.1.1.2

Cancel the common factor of $y$ .

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Step 3.1.1.2.1

Cancel the common factor.

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

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

Step 3.1.1.2.2

Divide $3 x$ by $1$ .

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

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

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

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

Step 3.1.1.3

Multiply $3$ by $- 1$ .

$2 y - 3 x = x y$

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

$2 y - 3 x = x y$

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

Step 3.1.2

Reorder $2 y$ and $- 3 x$ .

$- 3 x + 2 y = x y$

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

$- 3 x + 2 y = x y$

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

$- 3 x + 2 y = x y$

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

Step 4

Simplify the left side.

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Step 4.1

Simplify $\frac{4 x y}{x} + \frac{7 x y}{y}$ .

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Cancel the common factor of $x$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{4 x y}{x} + \frac{7 x y}{y} = x y$

$- 3 x + 2 y = x y$

Step 4.1.1.1.2

Divide $4 y$ by $1$ .

$4 y + \frac{7 x y}{y} = x y$

$- 3 x + 2 y = x y$

$4 y + \frac{7 x y}{y} = x y$

$- 3 x + 2 y = x y$

Step 4.1.1.2

Cancel the common factor of $y$ .

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Step 4.1.1.2.1

Cancel the common factor.

$4 y + \frac{7 x y}{y} = x y$

$- 3 x + 2 y = x y$

Step 4.1.1.2.2

Divide $7 x$ by $1$ .

$4 y + 7 x = x y$

$- 3 x + 2 y = x y$

$4 y + 7 x = x y$

$- 3 x + 2 y = x y$

$4 y + 7 x = x y$

$- 3 x + 2 y = x y$

Step 4.1.2

Reorder $4 y$ and $7 x$ .

$7 x + 4 y = x y$

$- 3 x + 2 y = x y$

$7 x + 4 y = x y$

$- 3 x + 2 y = x y$

$7 x + 4 y = x y$

$- 3 x + 2 y = x y$

Step 5

Move all terms containing variables to the left.

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Step 5.1

Subtract $x y$ from both sides of the equation.

$- 3 x + 2 y - x y = 0, 7 x + 4 y = x y$

Step 5.2

Subtract $x y$ from both sides of the equation.

$- 3 x + 2 y - x y = 0, 7 x + 4 y - x y = 0$

Step 6

Reorder the polynomial.

$- x y - 3 x + 2 y = 0$

$7 x + 4 y - x y = 0$

Step 7

Reorder the polynomial.

$- x y + 7 x + 4 y = 0$

$- x y - 3 x + 2 y = 0$

Step 8

Multiply each equation by the value that makes the coefficients of $y$ opposite.

$(- 1) \cdot (- x y - 3 x + 2 y) = (- 1) (0)$

$- x y + 7 x + 4 y = 0$

Step 9

Simplify.

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Step 9.1

Simplify the left side.

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Step 9.1.1

Simplify $(- 1) \cdot (- x y - 3 x + 2 y)$ .

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Step 9.1.1.1

Apply the distributive property.

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

$- x y + 7 x + 4 y = 0$

Step 9.1.1.2

Simplify.

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Step 9.1.1.2.1

Multiply $- 1 (- x y)$ .

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Step 9.1.1.2.1.1

Multiply $- 1$ by $- 1$ .

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

$- x y + 7 x + 4 y = 0$

Step 9.1.1.2.1.2

Multiply $x$ by $1$ .

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

$- x y + 7 x + 4 y = 0$

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

$- x y + 7 x + 4 y = 0$

Step 9.1.1.2.2

Multiply $- 3$ by $- 1$ .

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

$- x y + 7 x + 4 y = 0$

Step 9.1.1.2.3

Multiply $2$ by $- 1$ .

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

$- x y + 7 x + 4 y = 0$

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

$- x y + 7 x + 4 y = 0$

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

$- x y + 7 x + 4 y = 0$

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

$- x y + 7 x + 4 y = 0$

Step 9.2

Simplify the right side.

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Step 9.2.1

Multiply $- 1$ by $0$ .

$x y + 3 x - 2 y = 0$

$- x y + 7 x + 4 y = 0$

$x y + 3 x - 2 y = 0$

$- x y + 7 x + 4 y = 0$

$x y + 3 x - 2 y = 0$

$- x y + 7 x + 4 y = 0$

Step 10

Add the two equations together to eliminate $y$ from the system.

$x$

$y$

$x$

$y$

$+$

$3$

$x$

$+$

$3$

$x$

$-$

$2$

$y$

$-$

$2$

$y$

$=$

$0$

$+$

$-$

$x$

$y$

$+$

$7$

$x$

$-$

$x$

$y$

$+$

$7$

$x$

$-$

$x$

$y$

$+$

$4$

$y$

$=$

$0$

$1$

$0$

$x$

$+$

$2$

$y$

$=$

$0$

Step 11

Simplify the equation and solve for $x$ .

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Step 11.1

Subtract $2 y$ from both sides of the equation.

$10 x = - 2 y$

Step 11.2

Divide each term in $10 x = - 2 y$ by $10$ and simplify.

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Step 11.2.1

Divide each term in $10 x = - 2 y$ by $10$ .

$\frac{10 x}{10} = \frac{- 2 y}{10}$

Step 11.2.2

Simplify the left side.

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Step 11.2.2.1

Cancel the common factor of $10$ .

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Step 11.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{- 2 y}{10}$

Step 11.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2 y}{10}$

Step 11.2.3

Simplify the right side.

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Step 11.2.3.1

Cancel the common factor of $- 2$ and $10$ .

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Step 11.2.3.1.1

Factor $2$ out of $- 2 y$ .

$x = \frac{2 (- y)}{10}$

Step 11.2.3.1.2

Cancel the common factors.

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Step 11.2.3.1.2.1

Factor $2$ out of $10$ .

$x = \frac{2 (- y)}{2 (5)}$

Step 11.2.3.1.2.2

Cancel the common factor.

$x = \frac{2 (- y)}{2 \cdot 5}$

Step 11.2.3.1.2.3

Rewrite the expression.

$x = \frac{- y}{5}$

Step 11.2.3.2

Move the negative in front of the fraction.

$x = - \frac{y}{5}$

Step 12

Substitute the value found for $x$ into one of the original equations, then solve for $y$ .

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Step 12.1

Substitute the value found for $x$ into one of the original equations to solve for $y$ .

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

Step 12.2

Simplify $(- \frac{y}{5}) \cdot y + 3 (- \frac{y}{5}) - 2 y$ .

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Step 12.2.1

Simplify each term.

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Step 12.2.1.1

Multiply $(- \frac{y}{5}) y$ .

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Step 12.2.1.1.1

Combine $y$ and $\frac{y}{5}$ .

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

Step 12.2.1.1.2

Raise $y$ to the power of $1$ .

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

Step 12.2.1.1.3

Raise $y$ to the power of $1$ .

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

Step 12.2.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 12.2.1.1.5

Add $1$ and $1$ .

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

Step 12.2.1.2

Multiply $3 (- \frac{y}{5})$ .

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Step 12.2.1.2.1

Multiply $- 1$ by $3$ .

$- \frac{y^{2}}{5} - 3 \frac{y}{5} - 2 y = 0$

Step 12.2.1.2.2

Combine $- 3$ and $\frac{y}{5}$ .

$- \frac{y^{2}}{5} + \frac{- 3 y}{5} - 2 y = 0$

Step 12.2.1.3

Move the negative in front of the fraction.

$- \frac{y^{2}}{5} - \frac{3 y}{5} - 2 y = 0$

Step 12.2.2

To write $- 2 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$- \frac{y^{2}}{5} - \frac{3 y}{5} - 2 y \cdot \frac{5}{5} = 0$

Step 12.2.3

Combine $- 2 y$ and $\frac{5}{5}$ .

$- \frac{y^{2}}{5} - \frac{3 y}{5} + \frac{- 2 y \cdot 5}{5} = 0$

Step 12.2.4

Combine the numerators over the common denominator.

$- \frac{y^{2}}{5} + \frac{- 3 y - 2 y \cdot 5}{5} = 0$

Step 12.2.5

Combine the numerators over the common denominator.

$\frac{- y^{2} - 3 y - 2 y \cdot 5}{5} = 0$

Step 12.2.6

Multiply $5$ by $- 2$ .

$\frac{- y^{2} - 3 y - 10 y}{5} = 0$

Step 12.2.7

Subtract $10 y$ from $- 3 y$ .

$\frac{- y^{2} - 13 y}{5} = 0$

Step 12.2.8

Factor $y$ out of $- y^{2} - 13 y$ .

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Step 12.2.8.1

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

$\frac{y (- y) - 13 y}{5} = 0$

Step 12.2.8.2

Factor $y$ out of $- 13 y$ .

$\frac{y (- y) + y \cdot - 13}{5} = 0$

Step 12.2.8.3

Factor $y$ out of $y (- y) + y \cdot - 13$ .

$\frac{y (- y - 13)}{5} = 0$

Step 12.2.9

Factor $- 1$ out of $- y$ .

$\frac{y (- (y) - 13)}{5} = 0$

Step 12.2.10

Rewrite $- 13$ as $- 1 (13)$ .

$\frac{y (- (y) - 1 (13))}{5} = 0$

Step 12.2.11

Factor $- 1$ out of $- (y) - 1 (13)$ .

$\frac{y (- (y + 13))}{5} = 0$

Step 12.2.12

Simplify the expression.

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Step 12.2.12.1

Rewrite $- (y + 13)$ as $- 1 (y + 13)$ .

$\frac{y (- 1 (y + 13))}{5} = 0$

Step 12.2.12.2

Move the negative in front of the fraction.

$- \frac{y (y + 13)}{5} = 0$

Step 12.3

Set the numerator equal to zero.

$y (y + 13) = 0$

Step 12.4

Solve the equation for $y$ .

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Step 12.4.1

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

$y = 0$

$y + 13 = 0$

Step 12.4.2

Set $y$ equal to $0$ .

$y = 0$

Step 12.4.3

Set $y + 13$ equal to $0$ and solve for $y$ .

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Step 12.4.3.1

Set $y + 13$ equal to $0$ .

$y + 13 = 0$

Step 12.4.3.2

Subtract $13$ from both sides of the equation.

$y = - 13$

Step 12.4.4

The final solution is all the values that make $y (y + 13) = 0$ true.

$y = 0, - 13$

Step 13

The solution to the independent system of equations can be represented as a point.

$(- \frac{y}{5}, 0, - 13)$

Step 14

The result can be shown in multiple forms.

Point Form:

$(- \frac{y}{5}, 0, - 13)$

Equation Form:

$x = - \frac{y}{5}, y = 0, z = - 13$

Step 15