Determine if Dependent, Independent, or Inconsistent x+2y=2 , x/2+y=1

$x + 2 y = 2$ , $\frac{x}{2} + y = 1$

Step 1

Solve the system of equations.

Multiply each term in $\frac{x}{2} + y = 1$ by $2$ to eliminate the fractions.

Multiply each term in $\frac{x}{2} + y = 1$ by $2$ .

$\frac{x}{2} \cdot 2 + y \cdot 2 = 1 \cdot 2$

$x + 2 y = 2$

Simplify the left side.

Simplify each term.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x}{2} \cdot 2 + y \cdot 2 = 1 \cdot 2$

$x + 2 y = 2$

Rewrite the expression.

$x + y \cdot 2 = 1 \cdot 2$

$x + 2 y = 2$

$x + y \cdot 2 = 1 \cdot 2$

$x + 2 y = 2$

Move $2$ to the left of $y$ .

$x + 2 y = 1 \cdot 2$

$x + 2 y = 2$

$x + 2 y = 1 \cdot 2$

$x + 2 y = 2$

$x + 2 y = 1 \cdot 2$

$x + 2 y = 2$

Simplify the right side.

Multiply $2$ by $1$ .

$x + 2 y = 2$

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

$x + 2 y = 2$

$(- 1) \cdot (x + 2 y) = (- 1) (2)$

Simplify.

Simplify the left side.

Simplify $(- 1) \cdot (x + 2 y)$ .

Apply the distributive property.

$x + 2 y = 2$

$- 1 x - 1 (2 y) = (- 1) (2)$

Simplify the expression.

Rewrite $- 1 x$ as $- x$ .

$x + 2 y = 2$

$- x - 1 (2 y) = (- 1) (2)$

Multiply $2$ by $- 1$ .

$x + 2 y = 2$

$- x - 2 y = (- 1) (2)$

$x + 2 y = 2$

$- x - 2 y = (- 1) (2)$

$x + 2 y = 2$

$- x - 2 y = (- 1) (2)$

$x + 2 y = 2$

$- x - 2 y = (- 1) (2)$

Simplify the right side.

Multiply $- 1$ by $2$ .

$x + 2 y = 2$

$- x - 2 y = - 2$

$x + 2 y = 2$

$- x - 2 y = - 2$

$x + 2 y = 2$

$- x - 2 y = - 2$

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

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

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

Infinite number of solutions

Solve one of the equations for $y$ .

Subtract $x$ from both sides of the equation.

$2 y = 2 - x$

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Simplify each term.

Divide $2$ by $2$ .

$y = 1 + \frac{- x}{2}$

Move the negative in front of the fraction.

$y = 1 - \frac{x}{2}$

The solution is the set of ordered pairs that make $y = 1 - \frac{x}{2}$ true.

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

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 x+2y=2 , x/2+y=1

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