Algebra Examples

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

Step 1

Solve the system of equations.

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Step 1.1

Move all terms containing variables to the left.

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Step 1.1.1

Subtract $3 x$ from both sides of the equation.

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

Step 1.1.2

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

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

Step 1.2

Reorder the polynomial.

$- 3 x + y = 0$

$y + 3 x = 2$

Step 1.3

Reorder the polynomial.

$3 x + y = 2$

$- 3 x + y = 0$

Step 1.4

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

$- 3 x + y = 0$

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

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

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Step 1.5.1.1.1

Apply the distributive property.

$- 3 x + y = 0$

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

Step 1.5.1.1.2

Simplify the expression.

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Step 1.5.1.1.2.1

Multiply $3$ by $- 1$ .

$- 3 x + y = 0$

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

Step 1.5.1.1.2.2

Rewrite $- 1 y$ as $- y$ .

$- 3 x + y = 0$

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

$- 3 x + y = 0$

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

$- 3 x + y = 0$

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

$- 3 x + y = 0$

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

Step 1.5.2

Simplify the right side.

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Step 1.5.2.1

Multiply $- 1$ by $2$ .

$- 3 x + y = 0$

$- 3 x - y = - 2$

$- 3 x + y = 0$

$- 3 x - y = - 2$

$- 3 x + y = 0$

$- 3 x - y = - 2$

Step 1.6

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

	$-$	$3$	$x$	$+$	$y$	$=$		$0$
$+$	$-$	$3$	$x$	$-$	$y$	$=$	$-$	$2$
	$-$	$6$	$x$			$=$	$-$	$2$

Step 1.7

Divide each term in $- 6 x = - 2$ by $- 6$ and simplify.

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Step 1.7.1

Divide each term in $- 6 x = - 2$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- 2}{- 6}$

Step 1.7.2

Simplify the left side.

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Step 1.7.2.1

Cancel the common factor of $- 6$ .

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Step 1.7.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- 2}{- 6}$

Step 1.7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 2}{- 6}$

Step 1.7.3

Simplify the right side.

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Step 1.7.3.1

Cancel the common factor of $- 2$ and $- 6$ .

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Step 1.7.3.1.1

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

$x = \frac{- 2 (1)}{- 6}$

Step 1.7.3.1.2

Cancel the common factors.

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Step 1.7.3.1.2.1

Factor $- 2$ out of $- 6$ .

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

Step 1.7.3.1.2.2

Cancel the common factor.

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

Step 1.7.3.1.2.3

Rewrite the expression.

$x = \frac{1}{3}$

Step 1.8

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

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Step 1.8.1

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

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

Step 1.8.2

Cancel the common factor of $3$ .

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Step 1.8.2.1

Factor $3$ out of $- 3$ .

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

Step 1.8.2.2

Cancel the common factor.

$3 \cdot - 1 \frac{1}{3} + y = 0$

Step 1.8.2.3

Rewrite the expression.

$- 1 + y = 0$

Step 1.8.3

Add $1$ to both sides of the equation.

$y = 1$

Step 1.9

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

$(\frac{1}{3}, 1)$

Step 2

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

Independent

Step 3

	$-$	$3$	$x$	$+$	$y$	$=$		$0$
$+$	$-$	$3$	$x$	$-$	$y$	$=$	$-$	$2$
	$-$	$6$	$x$			$=$	$-$	$2$

	$-$	$3$	$x$	$+$	$y$	$=$		$0$
$+$	$-$	$3$	$x$	$-$	$y$	$=$	$-$	$2$
	$-$	$6$	$x$			$=$	$-$	$2$

	$-$	$3$	$x$	$+$	$y$	$=$		$0$
$+$	$-$	$3$	$x$	$-$	$y$	$=$	$-$	$2$
	$-$	$6$	$x$			$=$	$-$	$2$