Algebra Examples

$6 x + 8 y - 9 = 0$ , $6 y = 8 - 9 x$

Step 1

Solve the system of equations.

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Step 1.1

Simplify the right side.

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Step 1.1.1

Reorder $8$ and $- 9 x$ .

$6 y = - 9 x + 8$

$6 x + 8 y - 9 = 0$

$6 y = - 9 x + 8$

$6 x + 8 y - 9 = 0$

Step 1.2

Move all terms containing variables to the left.

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Step 1.2.1

Add $9$ to both sides of the equation.

$6 x + 8 y = 9, 6 y = - 9 x + 8$

Step 1.2.2

Add $9 x$ to both sides of the equation.

$6 x + 8 y = 9, 6 y + 9 x = 8$

Step 1.3

Reorder the polynomial.

$9 x + 6 y = 8$

$6 x + 8 y = 9$

Step 1.4

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

$(- 3) \cdot (6 x + 8 y) = (- 3) (9)$

$(2) \cdot (9 x + 6 y) = (2) (8)$

Step 1.5

Simplify.

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Step 1.5.1

Simplify the left side.

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Step 1.5.1.1

Simplify $(- 3) \cdot (6 x + 8 y)$ .

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Step 1.5.1.1.1

Apply the distributive property.

$- 3 (6 x) - 3 (8 y) = (- 3) (9)$

$(2) \cdot (9 x + 6 y) = (2) (8)$

Step 1.5.1.1.2

Multiply.

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Step 1.5.1.1.2.1

Multiply $6$ by $- 3$ .

$- 18 x - 3 (8 y) = (- 3) (9)$

$(2) \cdot (9 x + 6 y) = (2) (8)$

Step 1.5.1.1.2.2

Multiply $8$ by $- 3$ .

$- 18 x - 24 y = (- 3) (9)$

$(2) \cdot (9 x + 6 y) = (2) (8)$

$- 18 x - 24 y = (- 3) (9)$

$(2) \cdot (9 x + 6 y) = (2) (8)$

$- 18 x - 24 y = (- 3) (9)$

$(2) \cdot (9 x + 6 y) = (2) (8)$

$- 18 x - 24 y = (- 3) (9)$

$(2) \cdot (9 x + 6 y) = (2) (8)$

Step 1.5.2

Simplify the right side.

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Step 1.5.2.1

Multiply $- 3$ by $9$ .

$- 18 x - 24 y = - 27$

$(2) \cdot (9 x + 6 y) = (2) (8)$

$- 18 x - 24 y = - 27$

$(2) \cdot (9 x + 6 y) = (2) (8)$

Step 1.5.3

Simplify the left side.

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Step 1.5.3.1

Simplify $(2) \cdot (9 x + 6 y)$ .

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Step 1.5.3.1.1

Apply the distributive property.

$- 18 x - 24 y = - 27$

$2 (9 x) + 2 (6 y) = (2) (8)$

Step 1.5.3.1.2

Multiply.

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Step 1.5.3.1.2.1

Multiply $9$ by $2$ .

$- 18 x - 24 y = - 27$

$18 x + 2 (6 y) = (2) (8)$

Step 1.5.3.1.2.2

Multiply $6$ by $2$ .

$- 18 x - 24 y = - 27$

$18 x + 12 y = (2) (8)$

$- 18 x - 24 y = - 27$

$18 x + 12 y = (2) (8)$

$- 18 x - 24 y = - 27$

$18 x + 12 y = (2) (8)$

$- 18 x - 24 y = - 27$

$18 x + 12 y = (2) (8)$

Step 1.5.4

Simplify the right side.

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Step 1.5.4.1

Multiply $2$ by $8$ .

$- 18 x - 24 y = - 27$

$18 x + 12 y = 16$

$- 18 x - 24 y = - 27$

$18 x + 12 y = 16$

$- 18 x - 24 y = - 27$

$18 x + 12 y = 16$

Step 1.6

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

	$-$	$1$	$8$	$x$	$-$	$2$	$4$	$y$	$=$	$-$	$2$	$7$
$+$		$1$	$8$	$x$	$+$	$1$	$2$	$y$	$=$		$1$	$6$
					$-$	$1$	$2$	$y$	$=$	$-$	$1$	$1$

Step 1.7

Divide each term in $- 12 y = - 11$ by $- 12$ and simplify.

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Step 1.7.1

Divide each term in $- 12 y = - 11$ by $- 12$ .

$\frac{- 12 y}{- 12} = \frac{- 11}{- 12}$

Step 1.7.2

Simplify the left side.

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Step 1.7.2.1

Cancel the common factor of $- 12$ .

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Step 1.7.2.1.1

Cancel the common factor.

$\frac{- 12 y}{- 12} = \frac{- 11}{- 12}$

Step 1.7.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 11}{- 12}$

Step 1.7.3

Simplify the right side.

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Step 1.7.3.1

Dividing two negative values results in a positive value.

$y = \frac{11}{12}$

Step 1.8

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

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Step 1.8.1

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

$- 18 x - 24 (\frac{11}{12}) = - 27$

Step 1.8.2

Simplify each term.

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Step 1.8.2.1

Cancel the common factor of $12$ .

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Step 1.8.2.1.1

Factor $12$ out of $- 24$ .

$- 18 x + 12 (- 2) \frac{11}{12} = - 27$

Step 1.8.2.1.2

Cancel the common factor.

$- 18 x + 12 \cdot - 2 \frac{11}{12} = - 27$

Step 1.8.2.1.3

Rewrite the expression.

$- 18 x - 2 \cdot 11 = - 27$

Step 1.8.2.2

Multiply $- 2$ by $11$ .

$- 18 x - 22 = - 27$

Step 1.8.3

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

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Step 1.8.3.1

Add $22$ to both sides of the equation.

$- 18 x = - 27 + 22$

Step 1.8.3.2

Add $- 27$ and $22$ .

$- 18 x = - 5$

Step 1.8.4

Divide each term in $- 18 x = - 5$ by $- 18$ and simplify.

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Step 1.8.4.1

Divide each term in $- 18 x = - 5$ by $- 18$ .

$\frac{- 18 x}{- 18} = \frac{- 5}{- 18}$

Step 1.8.4.2

Simplify the left side.

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Step 1.8.4.2.1

Cancel the common factor of $- 18$ .

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Step 1.8.4.2.1.1

Cancel the common factor.

$\frac{- 18 x}{- 18} = \frac{- 5}{- 18}$

Step 1.8.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{- 18}$

Step 1.8.4.3

Simplify the right side.

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Step 1.8.4.3.1

Dividing two negative values results in a positive value.

$x = \frac{5}{18}$

Step 1.9

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

$(\frac{5}{18}, \frac{11}{12})$

Step 2

Since the system has a point of intersection, the system is independent.

Independent

Step 3

	$-$	$1$	$8$	$x$	$-$	$2$	$4$	$y$	$=$	$-$	$2$	$7$
$+$		$1$	$8$	$x$	$+$	$1$	$2$	$y$	$=$		$1$	$6$
					$-$	$1$	$2$	$y$	$=$	$-$	$1$	$1$

	$-$	$1$	$8$	$x$	$-$	$2$	$4$	$y$	$=$	$-$	$2$	$7$
$+$		$1$	$8$	$x$	$+$	$1$	$2$	$y$	$=$		$1$	$6$
					$-$	$1$	$2$	$y$	$=$	$-$	$1$	$1$

	$-$	$1$	$8$	$x$	$-$	$2$	$4$	$y$	$=$	$-$	$2$	$7$
$+$		$1$	$8$	$x$	$+$	$1$	$2$	$y$	$=$		$1$	$6$
					$-$	$1$	$2$	$y$	$=$	$-$	$1$	$1$