Determine if Dependent, Independent, or Inconsistent 8x-5y=3 , 16x-10y=6

$8 x - 5 y = 3$ , $16 x - 10 y = 6$

Step 1

Solve the system of equations.

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

$(- 2) \cdot (8 x - 5 y) = (- 2) (3)$

$16 x - 10 y = 6$

Simplify.

Simplify the left side.

Simplify $(- 2) \cdot (8 x - 5 y)$ .

Apply the distributive property.

$- 2 (8 x) - 2 (- 5 y) = (- 2) (3)$

$16 x - 10 y = 6$

Multiply.

Multiply $8$ by $- 2$ .

$- 16 x - 2 (- 5 y) = (- 2) (3)$

$16 x - 10 y = 6$

Multiply $- 5$ by $- 2$ .

$- 16 x + 10 y = (- 2) (3)$

$16 x - 10 y = 6$

$- 16 x + 10 y = (- 2) (3)$

$16 x - 10 y = 6$

$- 16 x + 10 y = (- 2) (3)$

$16 x - 10 y = 6$

$- 16 x + 10 y = (- 2) (3)$

$16 x - 10 y = 6$

Simplify the right side.

Multiply $- 2$ by $3$ .

$- 16 x + 10 y = - 6$

$16 x - 10 y = 6$

$- 16 x + 10 y = - 6$

$16 x - 10 y = 6$

$- 16 x + 10 y = - 6$

$16 x - 10 y = 6$

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

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

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

Infinite number of solutions

Solve one of the equations for $y$ .

Add $16 x$ to both sides of the equation.

$10 y = - 6 + 16 x$

Divide each term in $10 y = - 6 + 16 x$ by $10$ and simplify.

Divide each term in $10 y = - 6 + 16 x$ by $10$ .

$\frac{10 y}{10} = \frac{- 6}{10} + \frac{16 x}{10}$

Simplify the left side.

Cancel the common factor of $10$ .

Cancel the common factor.

$\frac{10 y}{10} = \frac{- 6}{10} + \frac{16 x}{10}$

Divide $y$ by $1$ .

$y = \frac{- 6}{10} + \frac{16 x}{10}$

Simplify the right side.

Simplify each term.

Cancel the common factor of $- 6$ and $10$ .

Factor $2$ out of $- 6$ .

$y = \frac{2 (- 3)}{10} + \frac{16 x}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$y = \frac{2 \cdot - 3}{2 \cdot 5} + \frac{16 x}{10}$

Cancel the common factor.

$y = \frac{2 \cdot - 3}{2 \cdot 5} + \frac{16 x}{10}$

Rewrite the expression.

$y = \frac{- 3}{5} + \frac{16 x}{10}$

Move the negative in front of the fraction.

$y = - \frac{3}{5} + \frac{16 x}{10}$

Cancel the common factor of $16$ and $10$ .

Factor $2$ out of $16 x$ .

$y = - \frac{3}{5} + \frac{2 (8 x)}{10}$

Cancel the common factors.

Factor $2$ out of $10$ .

$y = - \frac{3}{5} + \frac{2 (8 x)}{2 (5)}$

Cancel the common factor.

$y = - \frac{3}{5} + \frac{2 (8 x)}{2 \cdot 5}$

Rewrite the expression.

$y = - \frac{3}{5} + \frac{8 x}{5}$

The solution is the set of ordered pairs that make $y = - \frac{3}{5} + \frac{8 x}{5}$ true.

$(x, - \frac{3}{5} + \frac{8 x}{5})$

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 8x-5y=3 , 16x-10y=6

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