Algebra Examples

$2 x + 5 x = 21$ , $x + 2 y = 6$

Step 1

Simplify the left side.

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Step 1.1

Add $2 x$ and $5 x$ .

$7 x = 21$

$x + 2 y = 6$

$7 x = 21$

$x + 2 y = 6$

Step 2

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

$7 x = 21$

$(- 7) \cdot (x + 2 y) = (- 7) (6)$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $(- 7) \cdot (x + 2 y)$ .

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Step 3.1.1.1

Apply the distributive property.

$7 x = 21$

$- 7 x - 7 (2 y) = (- 7) (6)$

Step 3.1.1.2

Multiply $2$ by $- 7$ .

$7 x = 21$

$- 7 x - 14 y = (- 7) (6)$

$7 x = 21$

$- 7 x - 14 y = (- 7) (6)$

$7 x = 21$

$- 7 x - 14 y = (- 7) (6)$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $- 7$ by $6$ .

$7 x = 21$

$- 7 x - 14 y = - 42$

$7 x = 21$

$- 7 x - 14 y = - 42$

$7 x = 21$

$- 7 x - 14 y = - 42$

Step 4

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

					$7$	$x$	$=$		$2$	$1$
			$+$	$-$	$7$	$x$	$=$	$-$	$4$	$2$
$-$	$1$	$4$	$y$				$=$	$-$	$2$	$1$

Step 5

Divide each term in $- 14 y = - 21$ by $- 14$ and simplify.

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Step 5.1

Divide each term in $- 14 y = - 21$ by $- 14$ .

$\frac{- 14 y}{- 14} = \frac{- 21}{- 14}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $- 14$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{- 14 y}{- 14} = \frac{- 21}{- 14}$

Step 5.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 21}{- 14}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Cancel the common factor of $- 21$ and $- 14$ .

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Step 5.3.1.1

Factor $- 7$ out of $- 21$ .

$y = \frac{- 7 (3)}{- 14}$

Step 5.3.1.2

Cancel the common factors.

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Step 5.3.1.2.1

Factor $- 7$ out of $- 14$ .

$y = \frac{- 7 \cdot 3}{- 7 \cdot 2}$

Step 5.3.1.2.2

Cancel the common factor.

$y = \frac{- 7 \cdot 3}{- 7 \cdot 2}$

Step 5.3.1.2.3

Rewrite the expression.

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

Step 6

Substitute the value found for $y$ into one of the original equations, then solve for $x$ .

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Step 6.1

Substitute the value found for $y$ into one of the original equations to solve for $x$ .

$- 7 x - 14 (\frac{3}{2}) = - 42$

Step 6.2

Simplify each term.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Factor $2$ out of $- 14$ .

$- 7 x + 2 (- 7) \frac{3}{2} = - 42$

Step 6.2.1.2

Cancel the common factor.

$- 7 x + 2 \cdot - 7 \frac{3}{2} = - 42$

Step 6.2.1.3

Rewrite the expression.

$- 7 x - 7 \cdot 3 = - 42$

Step 6.2.2

Multiply $- 7$ by $3$ .

$- 7 x - 21 = - 42$

Step 6.3

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

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Step 6.3.1

Add $21$ to both sides of the equation.

$- 7 x = - 42 + 21$

Step 6.3.2

Add $- 42$ and $21$ .

$- 7 x = - 21$

Step 6.4

Divide each term in $- 7 x = - 21$ by $- 7$ and simplify.

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Step 6.4.1

Divide each term in $- 7 x = - 21$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{- 21}{- 7}$

Step 6.4.2

Simplify the left side.

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Step 6.4.2.1

Cancel the common factor of $- 7$ .

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Step 6.4.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{- 21}{- 7}$

Step 6.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 21}{- 7}$

Step 6.4.3

Simplify the right side.

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Step 6.4.3.1

Divide $- 21$ by $- 7$ .

$x = 3$

Step 7

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

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

Step 8

The result can be shown in multiple forms.

Point Form:

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

Equation Form:

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

Step 9

					$7$	$x$	$=$		$2$	$1$
			$+$	$-$	$7$	$x$	$=$	$-$	$4$	$2$
$-$	$1$	$4$	$y$				$=$	$-$	$2$	$1$

					$7$	$x$	$=$		$2$	$1$
			$+$	$-$	$7$	$x$	$=$	$-$	$4$	$2$
$-$	$1$	$4$	$y$				$=$	$-$	$2$	$1$

					$7$	$x$	$=$		$2$	$1$
			$+$	$-$	$7$	$x$	$=$	$-$	$4$	$2$
$-$	$1$	$4$	$y$				$=$	$-$	$2$	$1$