Algebra Examples

$2 x - 3 y = - 1$ $3 x + 4 y = 8$

Step 1

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

$(- 3) \cdot (2 x - 3 y) = (- 3) (- 1)$

$(2) \cdot (3 x + 4 y) = (2) (8)$

Step 2

Simplify.

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Step 2.1

Simplify the left side.

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Step 2.1.1

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

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Step 2.1.1.1

Apply the distributive property.

$- 3 (2 x) - 3 (- 3 y) = (- 3) (- 1)$

$(2) \cdot (3 x + 4 y) = (2) (8)$

Step 2.1.1.2

Multiply.

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Step 2.1.1.2.1

Multiply $2$ by $- 3$ .

$- 6 x - 3 (- 3 y) = (- 3) (- 1)$

$(2) \cdot (3 x + 4 y) = (2) (8)$

Step 2.1.1.2.2

Multiply $- 3$ by $- 3$ .

$- 6 x + 9 y = (- 3) (- 1)$

$(2) \cdot (3 x + 4 y) = (2) (8)$

$- 6 x + 9 y = (- 3) (- 1)$

$(2) \cdot (3 x + 4 y) = (2) (8)$

$- 6 x + 9 y = (- 3) (- 1)$

$(2) \cdot (3 x + 4 y) = (2) (8)$

$- 6 x + 9 y = (- 3) (- 1)$

$(2) \cdot (3 x + 4 y) = (2) (8)$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply $- 3$ by $- 1$ .

$- 6 x + 9 y = 3$

$(2) \cdot (3 x + 4 y) = (2) (8)$

$- 6 x + 9 y = 3$

$(2) \cdot (3 x + 4 y) = (2) (8)$

Step 2.3

Simplify the left side.

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Step 2.3.1

Simplify $(2) \cdot (3 x + 4 y)$ .

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Step 2.3.1.1

Apply the distributive property.

$- 6 x + 9 y = 3$

$2 (3 x) + 2 (4 y) = (2) (8)$

Step 2.3.1.2

Multiply.

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Step 2.3.1.2.1

Multiply $3$ by $2$ .

$- 6 x + 9 y = 3$

$6 x + 2 (4 y) = (2) (8)$

Step 2.3.1.2.2

Multiply $4$ by $2$ .

$- 6 x + 9 y = 3$

$6 x + 8 y = (2) (8)$

$- 6 x + 9 y = 3$

$6 x + 8 y = (2) (8)$

$- 6 x + 9 y = 3$

$6 x + 8 y = (2) (8)$

$- 6 x + 9 y = 3$

$6 x + 8 y = (2) (8)$

Step 2.4

Simplify the right side.

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Step 2.4.1

Multiply $2$ by $8$ .

$- 6 x + 9 y = 3$

$6 x + 8 y = 16$

$- 6 x + 9 y = 3$

$6 x + 8 y = 16$

$- 6 x + 9 y = 3$

$6 x + 8 y = 16$

Step 3

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

	$-$	$6$	$x$	$+$	$9$	$y$	$=$		$3$
$+$		$6$	$x$	$+$	$8$	$y$	$=$	$1$	$6$
				$1$	$7$	$y$	$=$	$1$	$9$

Step 4

Divide each term in $17 y = 19$ by $17$ and simplify.

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Step 4.1

Divide each term in $17 y = 19$ by $17$ .

$\frac{17 y}{17} = \frac{19}{17}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $17$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{17 y}{17} = \frac{19}{17}$

Step 4.2.1.2

Divide $y$ by $1$ .

$y = \frac{19}{17}$

Step 5

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

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Step 5.1

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

$- 6 x + 9 (\frac{19}{17}) = 3$

Step 5.2

Multiply $9 (\frac{19}{17})$ .

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Step 5.2.1

Combine $9$ and $\frac{19}{17}$ .

$- 6 x + \frac{9 \cdot 19}{17} = 3$

Step 5.2.2

Multiply $9$ by $19$ .

$- 6 x + \frac{171}{17} = 3$

Step 5.3

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

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Step 5.3.1

Subtract $\frac{171}{17}$ from both sides of the equation.

$- 6 x = 3 - \frac{171}{17}$

Step 5.3.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{17}{17}$ .

$- 6 x = 3 \cdot \frac{17}{17} - \frac{171}{17}$

Step 5.3.3

Combine $3$ and $\frac{17}{17}$ .

$- 6 x = \frac{3 \cdot 17}{17} - \frac{171}{17}$

Step 5.3.4

Combine the numerators over the common denominator.

$- 6 x = \frac{3 \cdot 17 - 171}{17}$

Step 5.3.5

Simplify the numerator.

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Step 5.3.5.1

Multiply $3$ by $17$ .

$- 6 x = \frac{51 - 171}{17}$

Step 5.3.5.2

Subtract $171$ from $51$ .

$- 6 x = \frac{- 120}{17}$

Step 5.3.6

Move the negative in front of the fraction.

$- 6 x = - \frac{120}{17}$

Step 5.4

Divide each term in $- 6 x = - \frac{120}{17}$ by $- 6$ and simplify.

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Step 5.4.1

Divide each term in $- 6 x = - \frac{120}{17}$ by $- 6$ .

$\frac{- 6 x}{- 6} = \frac{- \frac{120}{17}}{- 6}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $- 6$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{- 6 x}{- 6} = \frac{- \frac{120}{17}}{- 6}$

Step 5.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{120}{17}}{- 6}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{120}{17} \cdot \frac{1}{- 6}$

Step 5.4.3.2

Cancel the common factor of $6$ .

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Step 5.4.3.2.1

Move the leading negative in $- \frac{120}{17}$ into the numerator.

$x = \frac{- 120}{17} \cdot \frac{1}{- 6}$

Step 5.4.3.2.2

Factor $6$ out of $- 120$ .

$x = \frac{6 (- 20)}{17} \cdot \frac{1}{- 6}$

Step 5.4.3.2.3

Factor $6$ out of $- 6$ .

$x = \frac{6 \cdot - 20}{17} \cdot \frac{1}{6 \cdot - 1}$

Step 5.4.3.2.4

Cancel the common factor.

$x = \frac{6 \cdot - 20}{17} \cdot \frac{1}{6 \cdot - 1}$

Step 5.4.3.2.5

Rewrite the expression.

$x = \frac{- 20}{17} \cdot \frac{1}{- 1}$

Step 5.4.3.3

Multiply $\frac{- 20}{17}$ by $\frac{1}{- 1}$ .

$x = \frac{- 20}{17 \cdot - 1}$

Step 5.4.3.4

Multiply $17$ by $- 1$ .

$x = \frac{- 20}{- 17}$

Step 5.4.3.5

Dividing two negative values results in a positive value.

$x = \frac{20}{17}$

Step 6

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

$(\frac{20}{17}, \frac{19}{17})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(\frac{20}{17}, \frac{19}{17})$

Equation Form:

$x = \frac{20}{17}, y = \frac{19}{17}$

Step 8

	$-$	$6$	$x$	$+$	$9$	$y$	$=$		$3$
$+$		$6$	$x$	$+$	$8$	$y$	$=$	$1$	$6$
				$1$	$7$	$y$	$=$	$1$	$9$

	$-$	$6$	$x$	$+$	$9$	$y$	$=$		$3$
$+$		$6$	$x$	$+$	$8$	$y$	$=$	$1$	$6$
				$1$	$7$	$y$	$=$	$1$	$9$

	$-$	$6$	$x$	$+$	$9$	$y$	$=$		$3$
$+$		$6$	$x$	$+$	$8$	$y$	$=$	$1$	$6$
				$1$	$7$	$y$	$=$	$1$	$9$