Determine if Dependent, Independent, or Inconsistent 5x-8y=6 , 6x-6y=6

$5 x - 8 y = 6$ , $6 x - 6 y = 6$

Step 1

Solve the system of equations.

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

$(- 3) \cdot (5 x - 8 y) = (- 3) (6)$

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

Simplify.

Simplify the left side.

Simplify $(- 3) \cdot (5 x - 8 y)$ .

Apply the distributive property.

$- 3 (5 x) - 3 (- 8 y) = (- 3) (6)$

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

Multiply.

Multiply $5$ by $- 3$ .

$- 15 x - 3 (- 8 y) = (- 3) (6)$

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

Multiply $- 8$ by $- 3$ .

$- 15 x + 24 y = (- 3) (6)$

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

$- 15 x + 24 y = (- 3) (6)$

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

$- 15 x + 24 y = (- 3) (6)$

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

$- 15 x + 24 y = (- 3) (6)$

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

Simplify the right side.

Multiply $- 3$ by $6$ .

$- 15 x + 24 y = - 18$

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

$- 15 x + 24 y = - 18$

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

Simplify the left side.

Simplify $(4) \cdot (6 x - 6 y)$ .

Apply the distributive property.

$- 15 x + 24 y = - 18$

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

Multiply.

Multiply $6$ by $4$ .

$- 15 x + 24 y = - 18$

$24 x + 4 (- 6 y) = (4) (6)$

Multiply $- 6$ by $4$ .

$- 15 x + 24 y = - 18$

$24 x - 24 y = (4) (6)$

$- 15 x + 24 y = - 18$

$24 x - 24 y = (4) (6)$

$- 15 x + 24 y = - 18$

$24 x - 24 y = (4) (6)$

$- 15 x + 24 y = - 18$

$24 x - 24 y = (4) (6)$

Simplify the right side.

Multiply $4$ by $6$ .

$- 15 x + 24 y = - 18$

$24 x - 24 y = 24$

$- 15 x + 24 y = - 18$

$24 x - 24 y = 24$

$- 15 x + 24 y = - 18$

$24 x - 24 y = 24$

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

	$-$	$1$	$5$	$x$	$+$	$2$	$4$	$y$	$=$	$-$	$1$	$8$
$+$		$2$	$4$	$x$	$-$	$2$	$4$	$y$	$=$		$2$	$4$
			$9$	$x$					$=$			$6$

Divide each term in $9 x = 6$ by $9$ and simplify.

Divide each term in $9 x = 6$ by $9$ .

$\frac{9 x}{9} = \frac{6}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 x}{9} = \frac{6}{9}$

Divide $x$ by $1$ .

$x = \frac{6}{9}$

Simplify the right side.

Cancel the common factor of $6$ and $9$ .

Factor $3$ out of $6$ .

$x = \frac{3 (2)}{9}$

Cancel the common factors.

Factor $3$ out of $9$ .

$x = \frac{3 \cdot 2}{3 \cdot 3}$

Cancel the common factor.

$x = \frac{3 \cdot 2}{3 \cdot 3}$

Rewrite the expression.

$x = \frac{2}{3}$

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 $y$ .

$- 15 (\frac{2}{3}) + 24 y = - 18$

Simplify each term.

Cancel the common factor of $3$ .

Factor $3$ out of $- 15$ .

$3 (- 5) \frac{2}{3} + 24 y = - 18$

Cancel the common factor.

$3 \cdot - 5 \frac{2}{3} + 24 y = - 18$

Rewrite the expression.

$- 5 \cdot 2 + 24 y = - 18$

Multiply $- 5$ by $2$ .

$- 10 + 24 y = - 18$

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

Add $10$ to both sides of the equation.

$24 y = - 18 + 10$

Add $- 18$ and $10$ .

$24 y = - 8$

Divide each term in $24 y = - 8$ by $24$ and simplify.

Divide each term in $24 y = - 8$ by $24$ .

$\frac{24 y}{24} = \frac{- 8}{24}$

Simplify the left side.

Cancel the common factor of $24$ .

Cancel the common factor.

$\frac{24 y}{24} = \frac{- 8}{24}$

Divide $y$ by $1$ .

$y = \frac{- 8}{24}$

Simplify the right side.

Cancel the common factor of $- 8$ and $24$ .

Factor $8$ out of $- 8$ .

$y = \frac{8 (- 1)}{24}$

Cancel the common factors.

Factor $8$ out of $24$ .

$y = \frac{8 \cdot - 1}{8 \cdot 3}$

Cancel the common factor.

$y = \frac{8 \cdot - 1}{8 \cdot 3}$

Rewrite the expression.

$y = \frac{- 1}{3}$

Move the negative in front of the fraction.

$y = - \frac{1}{3}$

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

$(\frac{2}{3}, - \frac{1}{3})$

Step 2

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

Independent

Step 3

Determine if Dependent, Independent, or Inconsistent 5x-8y=6 , 6x-6y=6

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