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

$y = 4 x + 6$ $- 8 x + 2 y = 12$

Step 1

Solve the system of equations.

Subtract $4 x$ from both sides of the equation.

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

Reorder the polynomial.

$- 4 x + y = 6$

$- 8 x + 2 y = 12$

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

$(- 2) \cdot (- 4 x + y) = (- 2) (6)$

$- 8 x + 2 y = 12$

Simplify.

Simplify the left side.

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

Apply the distributive property.

$- 2 (- 4 x) - 2 y = (- 2) (6)$

$- 8 x + 2 y = 12$

Multiply $- 4$ by $- 2$ .

$8 x - 2 y = (- 2) (6)$

$- 8 x + 2 y = 12$

$8 x - 2 y = (- 2) (6)$

$- 8 x + 2 y = 12$

$8 x - 2 y = (- 2) (6)$

$- 8 x + 2 y = 12$

Simplify the right side.

Multiply $- 2$ by $6$ .

$8 x - 2 y = - 12$

$- 8 x + 2 y = 12$

$8 x - 2 y = - 12$

$- 8 x + 2 y = 12$

$8 x - 2 y = - 12$

$- 8 x + 2 y = 12$

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

		$8$	$x$	$-$	$2$	$y$	$=$	$-$	$1$	$2$
$+$	$-$	$8$	$x$	$+$	$2$	$y$	$=$		$1$	$2$
						$0$	$=$			$0$

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

Infinite number of solutions

Solve one of the equations for $y$ .

Subtract $8 x$ from both sides of the equation.

$- 2 y = - 12 - 8 x$

Divide each term in $- 2 y = - 12 - 8 x$ by $- 2$ and simplify.

Divide each term in $- 2 y = - 12 - 8 x$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{- 12}{- 2} + \frac{- 8 x}{- 2}$

Simplify the left side.

Cancel the common factor of $- 2$ .

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{- 12}{- 2} + \frac{- 8 x}{- 2}$

Divide $y$ by $1$ .

$y = \frac{- 12}{- 2} + \frac{- 8 x}{- 2}$

Simplify the right side.

Simplify each term.

Divide $- 12$ by $- 2$ .

$y = 6 + \frac{- 8 x}{- 2}$

Cancel the common factor of $- 8$ and $- 2$ .

Factor $- 2$ out of $- 8 x$ .

$y = 6 + \frac{- 2 (4 x)}{- 2}$

Cancel the common factors.

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

$y = 6 + \frac{- 2 (4 x)}{- 2 (1)}$

Cancel the common factor.

$y = 6 + \frac{- 2 (4 x)}{- 2 \cdot 1}$

Rewrite the expression.

$y = 6 + \frac{4 x}{1}$

Divide $4 x$ by $1$ .

$y = 6 + 4 x$

The solution is the set of ordered pairs that make $y = 6 + 4 x$ true.

$(x, 6 + 4 x)$

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 y=4x+6 -8x+2y=12

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