Determine if Dependent, Independent, or Inconsistent 3x+y=-5 3y+15=-9x

$3 x + y = - 5$ $3 y + 15 = - 9 x$

Step 1

Solve the system of equations.

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

Add $9 x$ to both sides of the equation.

$3 y + 15 + 9 x = 0$

$3 x + y = - 5$

Move $15$ .

$3 y + 9 x + 15 = 0$

$3 x + y = - 5$

Reorder $3 y$ and $9 x$ .

$9 x + 3 y + 15 = 0$

$3 x + y = - 5$

$9 x + 3 y + 15 = 0$

$3 x + y = - 5$

Subtract $15$ from both sides of the equation.

$9 x + 3 y = - 15$

$3 x + y = - 5$

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

$(- 3) \cdot (3 x + y) = (- 3) (- 5)$

$9 x + 3 y = - 15$

Simplify.

Simplify the left side.

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

Apply the distributive property.

$- 3 (3 x) - 3 y = (- 3) (- 5)$

$9 x + 3 y = - 15$

Multiply $3$ by $- 3$ .

$- 9 x - 3 y = (- 3) (- 5)$

$9 x + 3 y = - 15$

$- 9 x - 3 y = (- 3) (- 5)$

$9 x + 3 y = - 15$

$- 9 x - 3 y = (- 3) (- 5)$

$9 x + 3 y = - 15$

Simplify the right side.

Multiply $- 3$ by $- 5$ .

$- 9 x - 3 y = 15$

$9 x + 3 y = - 15$

$- 9 x - 3 y = 15$

$9 x + 3 y = - 15$

$- 9 x - 3 y = 15$

$9 x + 3 y = - 15$

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

	$-$	$9$	$x$	$-$	$3$	$y$	$=$		$1$	$5$
$+$		$9$	$x$	$+$	$3$	$y$	$=$	$-$	$1$	$5$
						$0$	$=$			$0$

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

Infinite number of solutions

Solve one of the equations for $y$ .

Add $9 x$ to both sides of the equation.

$- 3 y = 15 + 9 x$

Divide each term in $- 3 y = 15 + 9 x$ by $- 3$ and simplify.

Divide each term in $- 3 y = 15 + 9 x$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{15}{- 3} + \frac{9 x}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{15}{- 3} + \frac{9 x}{- 3}$

Divide $y$ by $1$ .

$y = \frac{15}{- 3} + \frac{9 x}{- 3}$

Simplify the right side.

Simplify each term.

Divide $15$ by $- 3$ .

$y = - 5 + \frac{9 x}{- 3}$

Cancel the common factor of $9$ and $- 3$ .

Factor $3$ out of $9 x$ .

$y = - 5 + \frac{3 (3 x)}{- 3}$

Move the negative one from the denominator of $\frac{3 x}{- 1}$ .

$y = - 5 - 1 \cdot (3 x)$

Rewrite $- 1 \cdot (3 x)$ as $- (3 x)$ .

$y = - 5 - (3 x)$

Multiply $3$ by $- 1$ .

$y = - 5 - 3 x$

The solution is the set of ordered pairs that make $y = - 5 - 3 x$ true.

$(x, - 5 - 3 x)$

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 3x+y=-5 3y+15=-9x

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