Algebra Examples

$y = - \frac{1}{2} x + 2$ $x + 2 y = 4$

Step 1

Solve the system of equations.

Tap for more steps...

Step 1.1

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

Tap for more steps...

Step 1.1.1

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

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

$x + 2 y = 4$

Step 1.1.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.1

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

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

$x + 2 y = 4$

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

$x + 2 y = 4$

Step 1.1.3

Simplify the right side.

Tap for more steps...

Step 1.1.3.1

Simplify each term.

Tap for more steps...

Step 1.1.3.1.1

Combine $x$ and $\frac{1}{2}$ .

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

$x + 2 y = 4$

Step 1.1.3.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.1.3.1.2.1

Move the leading negative in $- \frac{x}{2}$ into the numerator.

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

$x + 2 y = 4$

Step 1.1.3.1.2.2

Cancel the common factor.

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

$x + 2 y = 4$

Step 1.1.3.1.2.3

Rewrite the expression.

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

$x + 2 y = 4$

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

$x + 2 y = 4$

Step 1.1.3.1.3

Multiply $2$ by $2$ .

$2 y = - x + 4$

$x + 2 y = 4$

$2 y = - x + 4$

$x + 2 y = 4$

$2 y = - x + 4$

$x + 2 y = 4$

$2 y = - x + 4$

$x + 2 y = 4$

Step 1.2

Add $x$ to both sides of the equation.

$2 y + x = 4, x + 2 y = 4$

Step 1.3

Reorder the polynomial.

$x + 2 y = 4$

Step 1.4

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

$x + 2 y = 4$

$(- 1) \cdot (x + 2 y) = (- 1) (4)$

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

Simplify the left side.

Tap for more steps...

Step 1.5.1.1

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

Tap for more steps...

Step 1.5.1.1.1

Apply the distributive property.

$x + 2 y = 4$

$- 1 x - 1 (2 y) = (- 1) (4)$

Step 1.5.1.1.2

Simplify the expression.

Tap for more steps...

Step 1.5.1.1.2.1

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

$x + 2 y = 4$

$- x - 1 (2 y) = (- 1) (4)$

Step 1.5.1.1.2.2

Multiply $2$ by $- 1$ .

$x + 2 y = 4$

$- x - 2 y = (- 1) (4)$

$x + 2 y = 4$

$- x - 2 y = (- 1) (4)$

$x + 2 y = 4$

$- x - 2 y = (- 1) (4)$

$x + 2 y = 4$

$- x - 2 y = (- 1) (4)$

Step 1.5.2

Simplify the right side.

Tap for more steps...

Step 1.5.2.1

Multiply $- 1$ by $4$ .

$x + 2 y = 4$

$- x - 2 y = - 4$

$x + 2 y = 4$

$- x - 2 y = - 4$

$x + 2 y = 4$

$- x - 2 y = - 4$

Step 1.6

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

		$x$	$+$	$2$	$y$	$=$		$4$
$+$	$-$	$x$	$-$	$2$	$y$	$=$	$-$	$4$
					$0$	$=$		$0$

Step 1.7

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

Infinite number of solutions

Step 1.8

Solve one of the equations for $y$ .

Tap for more steps...

Step 1.8.1

Subtract $x$ from both sides of the equation.

$2 y = 4 - x$

Step 1.8.2

Divide each term in $2 y = 4 - x$ by $2$ and simplify.

Tap for more steps...

Step 1.8.2.1

Divide each term in $2 y = 4 - x$ by $2$ .

$\frac{2 y}{2} = \frac{4}{2} + \frac{- x}{2}$

Step 1.8.2.2

Simplify the left side.

Tap for more steps...

Step 1.8.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.8.2.2.1.1

Cancel the common factor.

$\frac{2 y}{2} = \frac{4}{2} + \frac{- x}{2}$

Step 1.8.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{4}{2} + \frac{- x}{2}$

Step 1.8.2.3

Simplify the right side.

Tap for more steps...

Step 1.8.2.3.1

Simplify each term.

Tap for more steps...

Step 1.8.2.3.1.1

Divide $4$ by $2$ .

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

Step 1.8.2.3.1.2

Move the negative in front of the fraction.

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

Step 1.9

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

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

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