Determine if Dependent, Independent, or Inconsistent x+2y=0 , 6y=-3x

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

Step 1

Solve the system of equations.

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

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

$6 y + 3 x = 0$

$x + 2 y = 0$

Reorder $6 y$ and $3 x$ .

$3 x + 6 y = 0$

$x + 2 y = 0$

$3 x + 6 y = 0$

$x + 2 y = 0$

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

$(- 3) \cdot (x + 2 y) = (- 3) (0)$

$3 x + 6 y = 0$

Simplify.

Simplify the left side.

Simplify $(- 3) \cdot (x + 2 y)$ .

Apply the distributive property.

$- 3 x - 3 (2 y) = (- 3) (0)$

$3 x + 6 y = 0$

Multiply $2$ by $- 3$ .

$- 3 x - 6 y = (- 3) (0)$

$3 x + 6 y = 0$

$- 3 x - 6 y = (- 3) (0)$

$3 x + 6 y = 0$

$- 3 x - 6 y = (- 3) (0)$

$3 x + 6 y = 0$

Simplify the right side.

Multiply $- 3$ by $0$ .

$- 3 x - 6 y = 0$

$3 x + 6 y = 0$

$- 3 x - 6 y = 0$

$3 x + 6 y = 0$

$- 3 x - 6 y = 0$

$3 x + 6 y = 0$

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

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

Since $0 = 0$ , the equations intersect at an infinite number of points.

Infinite number of solutions

Solve one of the equations for $y$ .

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

$- 6 y = 3 x$

Divide each term in $- 6 y = 3 x$ by $- 6$ and simplify.

Divide each term in $- 6 y = 3 x$ by $- 6$ .

$\frac{- 6 y}{- 6} = \frac{3 x}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 y}{- 6} = \frac{3 x}{- 6}$

Divide $y$ by $1$ .

$y = \frac{3 x}{- 6}$

Simplify the right side.

Cancel the common factor of $3$ and $- 6$ .

Factor $3$ out of $3 x$ .

$y = \frac{3 (x)}{- 6}$

Cancel the common factors.

Factor $3$ out of $- 6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

The solution is the set of ordered pairs that make $y = - \frac{x}{2}$ true.

$(x, - \frac{x}{2})$

Step 2

Since the system is always true, the equations are equal and the graphs are the same line. Thus, the system is dependent.

Dependent

Step 3

Determine if Dependent, Independent, or Inconsistent x+2y=0 , 6y=-3x

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