Algebra Examples

$3 x - 4 y = 13$ $y = - 3 x - 7$

Step 1

Solve the system of equations.

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Step 1.1

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

$3 x - 4 y = 13, y + 3 x = - 7$

Step 1.2

Reorder the polynomial.

$3 x + y = - 7$

$3 x - 4 y = 13$

Step 1.3

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

$(- 1) \cdot (3 x - 4 y) = (- 1) (13)$

$3 x + y = - 7$

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 $(- 1) \cdot (3 x - 4 y)$ .

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Step 1.4.1.1.1

Apply the distributive property.

$- 1 (3 x) - 1 (- 4 y) = (- 1) (13)$

$3 x + y = - 7$

Step 1.4.1.1.2

Multiply.

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Step 1.4.1.1.2.1

Multiply $3$ by $- 1$ .

$- 3 x - 1 (- 4 y) = (- 1) (13)$

$3 x + y = - 7$

Step 1.4.1.1.2.2

Multiply $- 4$ by $- 1$ .

$- 3 x + 4 y = (- 1) (13)$

$3 x + y = - 7$

$- 3 x + 4 y = (- 1) (13)$

$3 x + y = - 7$

$- 3 x + 4 y = (- 1) (13)$

$3 x + y = - 7$

$- 3 x + 4 y = (- 1) (13)$

$3 x + y = - 7$

Step 1.4.2

Simplify the right side.

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Step 1.4.2.1

Multiply $- 1$ by $13$ .

$- 3 x + 4 y = - 13$

$3 x + y = - 7$

$- 3 x + 4 y = - 13$

$3 x + y = - 7$

$- 3 x + 4 y = - 13$

$3 x + y = - 7$

Step 1.5

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

	$-$	$3$	$x$	$+$	$4$	$y$	$=$	$-$	$1$	$3$
$+$		$3$	$x$	$+$		$y$	$=$		$-$	$7$
					$5$	$y$	$=$	$-$	$2$	$0$

Step 1.6

Divide each term in $5 y = - 20$ by $5$ and simplify.

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Step 1.6.1

Divide each term in $5 y = - 20$ by $5$ .

$\frac{5 y}{5} = \frac{- 20}{5}$

Step 1.6.2

Simplify the left side.

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Step 1.6.2.1

Cancel the common factor of $5$ .

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Step 1.6.2.1.1

Cancel the common factor.

$\frac{5 y}{5} = \frac{- 20}{5}$

Step 1.6.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 20}{5}$

Step 1.6.3

Simplify the right side.

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Step 1.6.3.1

Divide $- 20$ by $5$ .

$y = - 4$

Step 1.7

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

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Step 1.7.1

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

$- 3 x + 4 (- 4) = - 13$

Step 1.7.2

Multiply $4$ by $- 4$ .

$- 3 x - 16 = - 13$

Step 1.7.3

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

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Step 1.7.3.1

Add $16$ to both sides of the equation.

$- 3 x = - 13 + 16$

Step 1.7.3.2

Add $- 13$ and $16$ .

$- 3 x = 3$

Step 1.7.4

Divide each term in $- 3 x = 3$ by $- 3$ and simplify.

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Step 1.7.4.1

Divide each term in $- 3 x = 3$ by $- 3$ .

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

Step 1.7.4.2

Simplify the left side.

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Step 1.7.4.2.1

Cancel the common factor of $- 3$ .

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Step 1.7.4.2.1.1

Cancel the common factor.

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

Step 1.7.4.2.1.2

Divide $x$ by $1$ .

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

Step 1.7.4.3

Simplify the right side.

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Step 1.7.4.3.1

Divide $3$ by $- 3$ .

$x = - 1$

Step 1.8

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

$(- 1, - 4)$

Step 2

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

Independent

Step 3

	$-$	$3$	$x$	$+$	$4$	$y$	$=$	$-$	$1$	$3$
$+$		$3$	$x$	$+$		$y$	$=$		$-$	$7$
					$5$	$y$	$=$	$-$	$2$	$0$

	$-$	$3$	$x$	$+$	$4$	$y$	$=$	$-$	$1$	$3$
$+$		$3$	$x$	$+$		$y$	$=$		$-$	$7$
					$5$	$y$	$=$	$-$	$2$	$0$

	$-$	$3$	$x$	$+$	$4$	$y$	$=$	$-$	$1$	$3$
$+$		$3$	$x$	$+$		$y$	$=$		$-$	$7$
					$5$	$y$	$=$	$-$	$2$	$0$