Algebra Examples

$5 x + 12 y - 11 = 0$ , $9 x = 8 - 4 y$

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 $- 4 y$ .

$9 x = - 4 y + 8$

$5 x + 12 y - 11 = 0$

$9 x = - 4 y + 8$

$5 x + 12 y - 11 = 0$

Step 1.2

Move all terms containing variables to the left.

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Step 1.2.1

Add $11$ to both sides of the equation.

$5 x + 12 y = 11, 9 x = - 4 y + 8$

Step 1.2.2

Add $4 y$ to both sides of the equation.

$5 x + 12 y = 11, 9 x + 4 y = 8$

Step 1.3

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

$5 x + 12 y = 11$

$(- 3) \cdot (9 x + 4 y) = (- 3) (8)$

Step 1.4

Simplify.

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Step 1.4.1

Simplify the left side.

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Step 1.4.1.1

Simplify $(- 3) \cdot (9 x + 4 y)$ .

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Step 1.4.1.1.1

Apply the distributive property.

$5 x + 12 y = 11$

$- 3 (9 x) - 3 (4 y) = (- 3) (8)$

Step 1.4.1.1.2

Multiply.

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Step 1.4.1.1.2.1

Multiply $9$ by $- 3$ .

$5 x + 12 y = 11$

$- 27 x - 3 (4 y) = (- 3) (8)$

Step 1.4.1.1.2.2

Multiply $4$ by $- 3$ .

$5 x + 12 y = 11$

$- 27 x - 12 y = (- 3) (8)$

$5 x + 12 y = 11$

$- 27 x - 12 y = (- 3) (8)$

$5 x + 12 y = 11$

$- 27 x - 12 y = (- 3) (8)$

$5 x + 12 y = 11$

$- 27 x - 12 y = (- 3) (8)$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Multiply $- 3$ by $8$ .

$5 x + 12 y = 11$

$- 27 x - 12 y = - 24$

$5 x + 12 y = 11$

$- 27 x - 12 y = - 24$

$5 x + 12 y = 11$

$- 27 x - 12 y = - 24$

Step 1.5

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

			$5$	$x$	$+$	$1$	$2$	$y$	$=$		$1$	$1$
$+$	$-$	$2$	$7$	$x$	$-$	$1$	$2$	$y$	$=$	$-$	$2$	$4$
	$-$	$2$	$2$	$x$					$=$	$-$	$1$	$3$

Step 1.6

Divide each term in $- 22 x = - 13$ by $- 22$ and simplify.

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Step 1.6.1

Divide each term in $- 22 x = - 13$ by $- 22$ .

$\frac{- 22 x}{- 22} = \frac{- 13}{- 22}$

Step 1.6.2

Simplify the left side.

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Step 1.6.2.1

Cancel the common factor of $- 22$ .

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Step 1.6.2.1.1

Cancel the common factor.

$\frac{- 22 x}{- 22} = \frac{- 13}{- 22}$

Step 1.6.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 13}{- 22}$

Step 1.6.3

Simplify the right side.

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Step 1.6.3.1

Dividing two negative values results in a positive value.

$x = \frac{13}{22}$

Step 1.7

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

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Step 1.7.1

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

$5 (\frac{13}{22}) + 12 y = 11$

Step 1.7.2

Multiply $5 (\frac{13}{22})$ .

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Step 1.7.2.1

Combine $5$ and $\frac{13}{22}$ .

$\frac{5 \cdot 13}{22} + 12 y = 11$

Step 1.7.2.2

Multiply $5$ by $13$ .

$\frac{65}{22} + 12 y = 11$

Step 1.7.3

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

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Step 1.7.3.1

Subtract $\frac{65}{22}$ from both sides of the equation.

$12 y = 11 - \frac{65}{22}$

Step 1.7.3.2

To write $11$ as a fraction with a common denominator, multiply by $\frac{22}{22}$ .

$12 y = 11 \cdot \frac{22}{22} - \frac{65}{22}$

Step 1.7.3.3

Combine $11$ and $\frac{22}{22}$ .

$12 y = \frac{11 \cdot 22}{22} - \frac{65}{22}$

Step 1.7.3.4

Combine the numerators over the common denominator.

$12 y = \frac{11 \cdot 22 - 65}{22}$

Step 1.7.3.5

Simplify the numerator.

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Step 1.7.3.5.1

Multiply $11$ by $22$ .

$12 y = \frac{242 - 65}{22}$

Step 1.7.3.5.2

Subtract $65$ from $242$ .

$12 y = \frac{177}{22}$

Step 1.7.4

Divide each term in $12 y = \frac{177}{22}$ by $12$ and simplify.

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Step 1.7.4.1

Divide each term in $12 y = \frac{177}{22}$ by $12$ .

$\frac{12 y}{12} = \frac{\frac{177}{22}}{12}$

Step 1.7.4.2

Simplify the left side.

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Step 1.7.4.2.1

Cancel the common factor of $12$ .

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Step 1.7.4.2.1.1

Cancel the common factor.

$\frac{12 y}{12} = \frac{\frac{177}{22}}{12}$

Step 1.7.4.2.1.2

Divide $y$ by $1$ .

$y = \frac{\frac{177}{22}}{12}$

Step 1.7.4.3

Simplify the right side.

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Step 1.7.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$y = \frac{177}{22} \cdot \frac{1}{12}$

Step 1.7.4.3.2

Cancel the common factor of $3$ .

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Step 1.7.4.3.2.1

Factor $3$ out of $177$ .

$y = \frac{3 (59)}{22} \cdot \frac{1}{12}$

Step 1.7.4.3.2.2

Factor $3$ out of $12$ .

$y = \frac{3 \cdot 59}{22} \cdot \frac{1}{3 \cdot 4}$

Step 1.7.4.3.2.3

Cancel the common factor.

$y = \frac{3 \cdot 59}{22} \cdot \frac{1}{3 \cdot 4}$

Step 1.7.4.3.2.4

Rewrite the expression.

$y = \frac{59}{22} \cdot \frac{1}{4}$

Step 1.7.4.3.3

Multiply $\frac{59}{22}$ by $\frac{1}{4}$ .

$y = \frac{59}{22 \cdot 4}$

Step 1.7.4.3.4

Multiply $22$ by $4$ .

$y = \frac{59}{88}$

Step 1.8

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

$(\frac{13}{22}, \frac{59}{88})$

Step 2

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

Independent

Step 3

			$5$	$x$	$+$	$1$	$2$	$y$	$=$		$1$	$1$
$+$	$-$	$2$	$7$	$x$	$-$	$1$	$2$	$y$	$=$	$-$	$2$	$4$
	$-$	$2$	$2$	$x$					$=$	$-$	$1$	$3$

			$5$	$x$	$+$	$1$	$2$	$y$	$=$		$1$	$1$
$+$	$-$	$2$	$7$	$x$	$-$	$1$	$2$	$y$	$=$	$-$	$2$	$4$
	$-$	$2$	$2$	$x$					$=$	$-$	$1$	$3$

			$5$	$x$	$+$	$1$	$2$	$y$	$=$		$1$	$1$
$+$	$-$	$2$	$7$	$x$	$-$	$1$	$2$	$y$	$=$	$-$	$2$	$4$
	$-$	$2$	$2$	$x$					$=$	$-$	$1$	$3$