Determine if Dependent, Independent, or Inconsistent 5x+y=-23 , 7x-3y=-41

$5 x + y = - 23$ , $7 x - 3 y = - 41$

Step 1

Solve the system of equations.

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

$(3) \cdot (5 x + y) = (3) (- 23)$

$7 x - 3 y = - 41$

Simplify.

Simplify the left side.

Simplify $(3) \cdot (5 x + y)$ .

Apply the distributive property.

$3 (5 x) + 3 y = (3) (- 23)$

$7 x - 3 y = - 41$

Multiply $5$ by $3$ .

$15 x + 3 y = (3) (- 23)$

$7 x - 3 y = - 41$

$15 x + 3 y = (3) (- 23)$

$7 x - 3 y = - 41$

$15 x + 3 y = (3) (- 23)$

$7 x - 3 y = - 41$

Simplify the right side.

Multiply $3$ by $- 23$ .

$15 x + 3 y = - 69$

$7 x - 3 y = - 41$

$15 x + 3 y = - 69$

$7 x - 3 y = - 41$

$15 x + 3 y = - 69$

$7 x - 3 y = - 41$

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

	$1$	$5$	$x$	$+$	$3$	$y$	$=$	$-$	$6$	$9$
$+$		$7$	$x$	$-$	$3$	$y$	$=$	$-$	$4$	$1$
	$2$	$2$	$x$				$=$	$-$	$1$	$1$	$0$

Divide each term in $22 x = - 110$ by $22$ and simplify.

Divide each term in $22 x = - 110$ by $22$ .

$\frac{22 x}{22} = \frac{- 110}{22}$

Simplify the left side.

Cancel the common factor of $22$ .

Cancel the common factor.

$\frac{22 x}{22} = \frac{- 110}{22}$

Divide $x$ by $1$ .

$x = \frac{- 110}{22}$

Simplify the right side.

Divide $- 110$ by $22$ .

$x = - 5$

Substitute the value found for $x$ into one of the original equations, then solve for $y$ .

Substitute the value found for $x$ into one of the original equations to solve for $y$ .

$15 (- 5) + 3 y = - 69$

Multiply $15$ by $- 5$ .

$- 75 + 3 y = - 69$

Move all terms not containing $y$ to the right side of the equation.

Add $75$ to both sides of the equation.

$3 y = - 69 + 75$

Add $- 69$ and $75$ .

$3 y = 6$

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Divide $6$ by $3$ .

$y = 2$

The solution to the independent system of equations can be represented as a point.

$(- 5, 2)$

Step 2

Since the system has a point of intersection, the system is independent.

Independent

Step 3

Determine if Dependent, Independent, or Inconsistent 5x+y=-23 , 7x-3y=-41

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