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

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

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.

$y - 3 x = 6$

$6 x - 2 y = - 12$

Reorder $y$ and $- 3 x$ .

$- 3 x + y = 6$

$6 x - 2 y = - 12$

$- 3 x + y = 6$

$6 x - 2 y = - 12$

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

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

$6 x - 2 y = - 12$

Simplify.

Simplify the left side.

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

Apply the distributive property.

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

$6 x - 2 y = - 12$

Multiply $- 3$ by $2$ .

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

$6 x - 2 y = - 12$

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

$6 x - 2 y = - 12$

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

$6 x - 2 y = - 12$

Simplify the right side.

Multiply $2$ by $6$ .

$- 6 x + 2 y = 12$

$6 x - 2 y = - 12$

$- 6 x + 2 y = 12$

$6 x - 2 y = - 12$

$- 6 x + 2 y = 12$

$6 x - 2 y = - 12$

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

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

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.

$2 y = 12 + 6 x$

Divide each term in $2 y = 12 + 6 x$ by $2$ and simplify.

Divide each term in $2 y = 12 + 6 x$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Divide $12$ by $2$ .

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

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

Factor $2$ out of $6 x$ .

$y = 6 + \frac{2 (3 x)}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$y = 6 + \frac{2 (3 x)}{2 (1)}$

Cancel the common factor.

$y = 6 + \frac{2 (3 x)}{2 \cdot 1}$

Rewrite the expression.

$y = 6 + \frac{3 x}{1}$

Divide $3 x$ by $1$ .

$y = 6 + 3 x$

The solution is the set of ordered pairs that make $y = 6 + 3 x$ true.

$(x, 6 + 3 x)$

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

Dependent

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

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