Determine if Dependent, Independent, or Inconsistent y=3x y=-3x+2

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

Step 1

Solve the system of equations.

Move all terms containing variables to the left side of the equation.

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

$y - 3 x = 0$

$y = - 3 x + 2$

Reorder $y$ and $- 3 x$ .

$- 3 x + y = 0$

$y = - 3 x + 2$

$- 3 x + y = 0$

$y = - 3 x + 2$

Move all terms containing variables to the left side of the equation.

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

$y + 3 x = 2$

$- 3 x + y = 0$

Reorder $y$ and $3 x$ .

$3 x + y = 2$

$- 3 x + y = 0$

$3 x + y = 2$

$- 3 x + y = 0$

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

$- 3 x + y = 0$

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

Simplify.

Simplify the left side.

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

Apply the distributive property.

$- 3 x + y = 0$

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

Simplify the expression.

Multiply $3$ by $- 1$ .

$- 3 x + y = 0$

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

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

$- 3 x + y = 0$

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

$- 3 x + y = 0$

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

$- 3 x + y = 0$

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

$- 3 x + y = 0$

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

Simplify the right side.

Multiply $- 1$ by $2$ .

$- 3 x + y = 0$

$- 3 x - y = - 2$

$- 3 x + y = 0$

$- 3 x - y = - 2$

$- 3 x + y = 0$

$- 3 x - y = - 2$

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

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

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

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

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

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

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

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

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

Cancel the common factors.

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

Cancel the common factor of $3$ .

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

$- 1 + y = 0$

Add $1$ to both sides of the equation.

$y = 1$

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

Determine if Dependent, Independent, or Inconsistent y=3x y=-3x+2

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