Determine if Dependent, Independent, or Inconsistent 2x-4y=8 , 3x-6y=12

$2 x - 4 y = 8$ , $3 x - 6 y = 12$

Step 1

Solve the system of equations.

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

$(- 3) \cdot (2 x - 4 y) = (- 3) (8)$

$(2) \cdot (3 x - 6 y) = (2) (12)$

Simplify.

Simplify the left side.

Simplify $(- 3) \cdot (2 x - 4 y)$ .

Apply the distributive property.

$- 3 (2 x) - 3 (- 4 y) = (- 3) (8)$

$(2) \cdot (3 x - 6 y) = (2) (12)$

Multiply.

Multiply $2$ by $- 3$ .

$- 6 x - 3 (- 4 y) = (- 3) (8)$

$(2) \cdot (3 x - 6 y) = (2) (12)$

Multiply $- 4$ by $- 3$ .

$- 6 x + 12 y = (- 3) (8)$

$(2) \cdot (3 x - 6 y) = (2) (12)$

$- 6 x + 12 y = (- 3) (8)$

$(2) \cdot (3 x - 6 y) = (2) (12)$

$- 6 x + 12 y = (- 3) (8)$

$(2) \cdot (3 x - 6 y) = (2) (12)$

$- 6 x + 12 y = (- 3) (8)$

$(2) \cdot (3 x - 6 y) = (2) (12)$

Simplify the right side.

Multiply $- 3$ by $8$ .

$- 6 x + 12 y = - 24$

$(2) \cdot (3 x - 6 y) = (2) (12)$

$- 6 x + 12 y = - 24$

$(2) \cdot (3 x - 6 y) = (2) (12)$

Simplify the left side.

Simplify $(2) \cdot (3 x - 6 y)$ .

Apply the distributive property.

$- 6 x + 12 y = - 24$

$2 (3 x) + 2 (- 6 y) = (2) (12)$

Multiply.

Multiply $3$ by $2$ .

$- 6 x + 12 y = - 24$

$6 x + 2 (- 6 y) = (2) (12)$

Multiply $- 6$ by $2$ .

$- 6 x + 12 y = - 24$

$6 x - 12 y = (2) (12)$

$- 6 x + 12 y = - 24$

$6 x - 12 y = (2) (12)$

$- 6 x + 12 y = - 24$

$6 x - 12 y = (2) (12)$

$- 6 x + 12 y = - 24$

$6 x - 12 y = (2) (12)$

Simplify the right side.

Multiply $2$ by $12$ .

$- 6 x + 12 y = - 24$

$6 x - 12 y = 24$

$- 6 x + 12 y = - 24$

$6 x - 12 y = 24$

$- 6 x + 12 y = - 24$

$6 x - 12 y = 24$

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

	$-$	$6$	$x$	$+$	$1$	$2$	$y$	$=$	$-$	$2$	$4$
$+$		$6$	$x$	$-$	$1$	$2$	$y$	$=$		$2$	$4$
							$0$	$=$			$0$

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

Infinite number of solutions

Solve one of the equations for $y$ .

Add $6 x$ to both sides of the equation.

$12 y = - 24 + 6 x$

Divide each term in $12 y = - 24 + 6 x$ by $12$ and simplify.

Divide each term in $12 y = - 24 + 6 x$ by $12$ .

$\frac{12 y}{12} = \frac{- 24}{12} + \frac{6 x}{12}$

Simplify the left side.

Cancel the common factor of $12$ .

Cancel the common factor.

$\frac{12 y}{12} = \frac{- 24}{12} + \frac{6 x}{12}$

Divide $y$ by $1$ .

$y = \frac{- 24}{12} + \frac{6 x}{12}$

Simplify the right side.

Simplify each term.

Divide $- 24$ by $12$ .

$y = - 2 + \frac{6 x}{12}$

Cancel the common factor of $6$ and $12$ .

Factor $6$ out of $6 x$ .

$y = - 2 + \frac{6 (x)}{12}$

Cancel the common factors.

Factor $6$ out of $12$ .

$y = - 2 + \frac{6 x}{6 \cdot 2}$

Cancel the common factor.

$y = - 2 + \frac{6 x}{6 \cdot 2}$

Rewrite the expression.

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

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

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

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 2x-4y=8 , 3x-6y=12

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