Algebra Examples

$2 x + 6 y = - 28$ $x + 3 y = - 14$

Step 1

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

$2 x + 6 y = - 28$

$(- 2) \cdot (x + 3 y) = (- 2) (- 14)$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

Simplify $(- 2) \cdot (x + 3 y)$ .

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Step 2.1.1.1

Apply the distributive property.

$2 x + 6 y = - 28$

$- 2 x - 2 (3 y) = (- 2) (- 14)$

Step 2.1.1.2

Multiply $3$ by $- 2$ .

$2 x + 6 y = - 28$

$- 2 x - 6 y = (- 2) (- 14)$

$2 x + 6 y = - 28$

$- 2 x - 6 y = (- 2) (- 14)$

$2 x + 6 y = - 28$

$- 2 x - 6 y = (- 2) (- 14)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $- 2$ by $- 14$ .

$2 x + 6 y = - 28$

$- 2 x - 6 y = 28$

$2 x + 6 y = - 28$

$- 2 x - 6 y = 28$

$2 x + 6 y = - 28$

$- 2 x - 6 y = 28$

Step 3

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

		$2$	$x$	$+$	$6$	$y$	$=$	$-$	$2$	$8$
$+$	$-$	$2$	$x$	$-$	$6$	$y$	$=$		$2$	$8$
						$0$	$=$			$0$

Step 4

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

Infinite number of solutions

Step 5

Solve one of the equations for $y$ .

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Step 5.1

Subtract $2 x$ from both sides of the equation.

$6 y = - 28 - 2 x$

Step 5.2

Divide each term in $6 y = - 28 - 2 x$ by $6$ and simplify.

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Step 5.2.1

Divide each term in $6 y = - 28 - 2 x$ by $6$ .

$\frac{6 y}{6} = \frac{- 28}{6} + \frac{- 2 x}{6}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $6$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{6 y}{6} = \frac{- 28}{6} + \frac{- 2 x}{6}$

Step 5.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 28}{6} + \frac{- 2 x}{6}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Cancel the common factor of $- 28$ and $6$ .

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Step 5.2.3.1.1.1

Factor $2$ out of $- 28$ .

$y = \frac{2 (- 14)}{6} + \frac{- 2 x}{6}$

Step 5.2.3.1.1.2

Cancel the common factors.

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Step 5.2.3.1.1.2.1

Factor $2$ out of $6$ .

$y = \frac{2 \cdot - 14}{2 \cdot 3} + \frac{- 2 x}{6}$

Step 5.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{2 \cdot - 14}{2 \cdot 3} + \frac{- 2 x}{6}$

Step 5.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{- 14}{3} + \frac{- 2 x}{6}$

Step 5.2.3.1.2

Move the negative in front of the fraction.

$y = - \frac{14}{3} + \frac{- 2 x}{6}$

Step 5.2.3.1.3

Cancel the common factor of $- 2$ and $6$ .

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Step 5.2.3.1.3.1

Factor $2$ out of $- 2 x$ .

$y = - \frac{14}{3} + \frac{2 (- x)}{6}$

Step 5.2.3.1.3.2

Cancel the common factors.

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Step 5.2.3.1.3.2.1

Factor $2$ out of $6$ .

$y = - \frac{14}{3} + \frac{2 (- x)}{2 (3)}$

Step 5.2.3.1.3.2.2

Cancel the common factor.

$y = - \frac{14}{3} + \frac{2 (- x)}{2 \cdot 3}$

Step 5.2.3.1.3.2.3

Rewrite the expression.

$y = - \frac{14}{3} + \frac{- x}{3}$

Step 5.2.3.1.4

Move the negative in front of the fraction.

$y = - \frac{14}{3} - \frac{x}{3}$

Step 6

The solution is the set of ordered pairs that make $y = - \frac{14}{3} - \frac{x}{3}$ true.

$(x, - \frac{14}{3} - \frac{x}{3})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(x, - \frac{14}{3} - \frac{x}{3})$

Equation Form:

$x = x, y = - \frac{14}{3} - \frac{x}{3}$

Step 8

		$2$	$x$	$+$	$6$	$y$	$=$	$-$	$2$	$8$
$+$	$-$	$2$	$x$	$-$	$6$	$y$	$=$		$2$	$8$
						$0$	$=$			$0$

		$2$	$x$	$+$	$6$	$y$	$=$	$-$	$2$	$8$
$+$	$-$	$2$	$x$	$-$	$6$	$y$	$=$		$2$	$8$
						$0$	$=$			$0$

		$2$	$x$	$+$	$6$	$y$	$=$	$-$	$2$	$8$
$+$	$-$	$2$	$x$	$-$	$6$	$y$	$=$		$2$	$8$
						$0$	$=$			$0$