Algebra Examples

$y = x + \frac{36}{36}$ , $108 x + 72 y = 102$

Step 1

Multiply each term in $y = x + \frac{36}{36}$ by $36$ to eliminate the fractions.

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Step 1.1

Multiply each term in $y = x + \frac{36}{36}$ by $36$ .

$y \cdot 36 = x \cdot 36 + \frac{36}{36} \cdot 36$

$108 x + 72 y = 102$

Step 1.2

Simplify the left side.

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Step 1.2.1

Move $36$ to the left of $y$ .

$36 y = x \cdot 36 + \frac{36}{36} \cdot 36$

$108 x + 72 y = 102$

$36 y = x \cdot 36 + \frac{36}{36} \cdot 36$

$108 x + 72 y = 102$

Step 1.3

Simplify the right side.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Move $36$ to the left of $x$ .

$36 y = 36 \cdot x + \frac{36}{36} \cdot 36$

$108 x + 72 y = 102$

Step 1.3.1.2

Cancel the common factor of $36$ .

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Step 1.3.1.2.1

Cancel the common factor.

$36 y = 36 x + \frac{36}{36} \cdot 36$

$108 x + 72 y = 102$

Step 1.3.1.2.2

Rewrite the expression.

$36 y = 36 x + 1 \cdot 36$

$108 x + 72 y = 102$

$36 y = 36 x + 1 \cdot 36$

$108 x + 72 y = 102$

Step 1.3.1.3

Multiply $36$ by $1$ .

$36 y = 36 x + 36$

$108 x + 72 y = 102$

$36 y = 36 x + 36$

$108 x + 72 y = 102$

$36 y = 36 x + 36$

$108 x + 72 y = 102$

$36 y = 36 x + 36$

$108 x + 72 y = 102$

Step 2

Subtract $36 x$ from both sides of the equation.

$36 y - 36 x = 36, 108 x + 72 y = 102$

Step 3

Reorder the polynomial.

$- 36 x + 36 y = 36$

$108 x + 72 y = 102$

Step 4

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

$(- 2) \cdot (- 36 x + 36 y) = (- 2) (36)$

$108 x + 72 y = 102$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $(- 2) \cdot (- 36 x + 36 y)$ .

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Step 5.1.1.1

Apply the distributive property.

$- 2 (- 36 x) - 2 (36 y) = (- 2) (36)$

$108 x + 72 y = 102$

Step 5.1.1.2

Multiply.

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Step 5.1.1.2.1

Multiply $- 36$ by $- 2$ .

$72 x - 2 (36 y) = (- 2) (36)$

$108 x + 72 y = 102$

Step 5.1.1.2.2

Multiply $36$ by $- 2$ .

$72 x - 72 y = (- 2) (36)$

$108 x + 72 y = 102$

$72 x - 72 y = (- 2) (36)$

$108 x + 72 y = 102$

$72 x - 72 y = (- 2) (36)$

$108 x + 72 y = 102$

$72 x - 72 y = (- 2) (36)$

$108 x + 72 y = 102$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $- 2$ by $36$ .

$72 x - 72 y = - 72$

$108 x + 72 y = 102$

$72 x - 72 y = - 72$

$108 x + 72 y = 102$

$72 x - 72 y = - 72$

$108 x + 72 y = 102$

Step 6

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

		$7$	$2$	$x$	$-$	$7$	$2$	$y$	$=$	$-$	$7$	$2$
$+$	$1$	$0$	$8$	$x$	$+$	$7$	$2$	$y$	$=$	$1$	$0$	$2$
	$1$	$8$	$0$	$x$					$=$		$3$	$0$

Step 7

Divide each term in $180 x = 30$ by $180$ and simplify.

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Step 7.1

Divide each term in $180 x = 30$ by $180$ .

$\frac{180 x}{180} = \frac{30}{180}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $180$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{180 x}{180} = \frac{30}{180}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{30}{180}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Cancel the common factor of $30$ and $180$ .

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Step 7.3.1.1

Factor $30$ out of $30$ .

$x = \frac{30 (1)}{180}$

Step 7.3.1.2

Cancel the common factors.

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Step 7.3.1.2.1

Factor $30$ out of $180$ .

$x = \frac{30 \cdot 1}{30 \cdot 6}$

Step 7.3.1.2.2

Cancel the common factor.

$x = \frac{30 \cdot 1}{30 \cdot 6}$

Step 7.3.1.2.3

Rewrite the expression.

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

Step 8

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

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Step 8.1

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

$72 (\frac{1}{6}) - 72 y = - 72$

Step 8.2

Cancel the common factor of $6$ .

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Step 8.2.1

Factor $6$ out of $72$ .

$6 (12) \frac{1}{6} - 72 y = - 72$

Step 8.2.2

Cancel the common factor.

$6 \cdot 12 \frac{1}{6} - 72 y = - 72$

Step 8.2.3

Rewrite the expression.

$12 - 72 y = - 72$

Step 8.3

Move all terms not containing $y$ to the right side of the equation.

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Step 8.3.1

Subtract $12$ from both sides of the equation.

$- 72 y = - 72 - 12$

Step 8.3.2

Subtract $12$ from $- 72$ .

$- 72 y = - 84$

Step 8.4

Divide each term in $- 72 y = - 84$ by $- 72$ and simplify.

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Step 8.4.1

Divide each term in $- 72 y = - 84$ by $- 72$ .

$\frac{- 72 y}{- 72} = \frac{- 84}{- 72}$

Step 8.4.2

Simplify the left side.

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Step 8.4.2.1

Cancel the common factor of $- 72$ .

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Step 8.4.2.1.1

Cancel the common factor.

$\frac{- 72 y}{- 72} = \frac{- 84}{- 72}$

Step 8.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 84}{- 72}$

Step 8.4.3

Simplify the right side.

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Step 8.4.3.1

Cancel the common factor of $- 84$ and $- 72$ .

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Step 8.4.3.1.1

Factor $- 12$ out of $- 84$ .

$y = \frac{- 12 (7)}{- 72}$

Step 8.4.3.1.2

Cancel the common factors.

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Step 8.4.3.1.2.1

Factor $- 12$ out of $- 72$ .

$y = \frac{- 12 \cdot 7}{- 12 \cdot 6}$

Step 8.4.3.1.2.2

Cancel the common factor.

$y = \frac{- 12 \cdot 7}{- 12 \cdot 6}$

Step 8.4.3.1.2.3

Rewrite the expression.

$y = \frac{7}{6}$

Step 9

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

$(\frac{1}{6}, \frac{7}{6})$

Step 10

The result can be shown in multiple forms.

Point Form:

$(\frac{1}{6}, \frac{7}{6})$

Equation Form:

$x = \frac{1}{6}, y = \frac{7}{6}$

Step 11

		$7$	$2$	$x$	$-$	$7$	$2$	$y$	$=$	$-$	$7$	$2$
$+$	$1$	$0$	$8$	$x$	$+$	$7$	$2$	$y$	$=$	$1$	$0$	$2$
	$1$	$8$	$0$	$x$					$=$		$3$	$0$

		$7$	$2$	$x$	$-$	$7$	$2$	$y$	$=$	$-$	$7$	$2$
$+$	$1$	$0$	$8$	$x$	$+$	$7$	$2$	$y$	$=$	$1$	$0$	$2$
	$1$	$8$	$0$	$x$					$=$		$3$	$0$

		$7$	$2$	$x$	$-$	$7$	$2$	$y$	$=$	$-$	$7$	$2$
$+$	$1$	$0$	$8$	$x$	$+$	$7$	$2$	$y$	$=$	$1$	$0$	$2$
	$1$	$8$	$0$	$x$					$=$		$3$	$0$