Algebra Examples

3 x + y = 4

$3 x + y = 4$ ,

6 x - 7 y = 2

$6 x - 7 y = 2$

Step 1

Multiply each equation by the value that makes the coefficients of

x

$x$ opposite.

(- 2) \cdot (3 x + y) = (- 2) (4)

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

6 x - 7 y = 2

$6 x - 7 y = 2$

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 (3 x + y)

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

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Step 2.1.1.1

Apply the distributive property.

- 2 (3 x) - 2 y = (- 2) (4)

$- 2 (3 x) - 2 y = (- 2) (4)$

6 x - 7 y = 2

$6 x - 7 y = 2$

Step 2.1.1.2

Multiply

3

$3$ by

- 2

$- 2$ .

- 6 x - 2 y = (- 2) (4)

$- 6 x - 2 y = (- 2) (4)$

6 x - 7 y = 2

$6 x - 7 y = 2$

- 6 x - 2 y = (- 2) (4)

$- 6 x - 2 y = (- 2) (4)$

6 x - 7 y = 2

$6 x - 7 y = 2$

- 6 x - 2 y = (- 2) (4)

$- 6 x - 2 y = (- 2) (4)$

6 x - 7 y = 2

$6 x - 7 y = 2$

Step 2.2

Simplify the right side.

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Step 2.2.1

Multiply

- 2

$- 2$ by

4

$4$ .

- 6 x - 2 y = - 8

$- 6 x - 2 y = - 8$

6 x - 7 y = 2

$6 x - 7 y = 2$

- 6 x - 2 y = - 8

$- 6 x - 2 y = - 8$

6 x - 7 y = 2

$6 x - 7 y = 2$

- 6 x - 2 y = - 8

$- 6 x - 2 y = - 8$

6 x - 7 y = 2

$6 x - 7 y = 2$

Step 3

Add the two equations together to eliminate

x

$x$ from the system.

	$-$ $-$	$6$ $6$	$x$ $x$	$-$ $-$	$2$ $2$	$y$ $y$	$=$ $=$	$-$ $-$	$8$ $8$
$+$ $+$		$6$ $6$	$x$ $x$	$-$ $-$	$7$ $7$	$y$ $y$	$=$ $=$		$2$ $2$
				$-$ $-$	$9$ $9$	$y$ $y$	$=$ $=$	$-$ $-$	$6$ $6$

Step 4

Divide each term in

- 9 y = - 6

$- 9 y = - 6$ by

- 9

$- 9$ and simplify.

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Step 4.1

Divide each term in

- 9 y = - 6

$- 9 y = - 6$ by

- 9

$- 9$ .

\frac{- 9 y}{- 9} = \frac{- 6}{- 9}

$\frac{- 9 y}{- 9} = \frac{- 6}{- 9}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of

- 9

$- 9$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 9 y}{- 9} = \frac{- 6}{- 9}$

Step 4.2.1.2

Divide

$y$ by

$1$ .

$y = \frac{- 6}{- 9}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of

$- 6$ and

$- 9$ .

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Step 4.3.1.1

Factor

$- 3$ out of

$- 6$ .

$y = \frac{- 3 (2)}{- 9}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor

$- 3$ out of

$- 9$ .

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

Step 4.3.1.2.2

Cancel the common factor.

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

Step 4.3.1.2.3

Rewrite the expression.

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

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

$x$ .

$- 6 x - 2 (\frac{2}{3}) = - 8$

Step 5.2

Simplify each term.

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Step 5.2.1

Multiply

$- 2 (\frac{2}{3})$ .

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Step 5.2.1.1

Combine

$- 2$ and

$\frac{2}{3}$ .

$- 6 x + \frac{- 2 \cdot 2}{3} = - 8$

Step 5.2.1.2

Multiply

$- 2$ by

$2$ .

$- 6 x + \frac{- 4}{3} = - 8$

Step 5.2.2

Move the negative in front of the fraction.

$- 6 x - \frac{4}{3} = - 8$

Step 5.3

Move all terms not containing

$x$ to the right side of the equation.

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Step 5.3.1

Add

$\frac{4}{3}$ to both sides of the equation.

$- 6 x = - 8 + \frac{4}{3}$

Step 5.3.2

To write

$- 8$ as a fraction with a common denominator, multiply by

$\frac{3}{3}$ .

$- 6 x = - 8 \cdot \frac{3}{3} + \frac{4}{3}$

Step 5.3.3

Combine

$- 8$ and

$\frac{3}{3}$ .

$- 6 x = \frac{- 8 \cdot 3}{3} + \frac{4}{3}$

Step 5.3.4

Combine the numerators over the common denominator.

$- 6 x = \frac{- 8 \cdot 3 + 4}{3}$

Step 5.3.5

Simplify the numerator.

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Step 5.3.5.1

Multiply

$- 8$ by

$3$ .

$- 6 x = \frac{- 24 + 4}{3}$

Step 5.3.5.2

Add

$- 24$ and

$4$ .

$- 6 x = \frac{- 20}{3}$

Step 5.3.6

Move the negative in front of the fraction.

$- 6 x = - \frac{20}{3}$

Step 5.4

Divide each term in

$- 6 x = - \frac{20}{3}$ by

$- 6$ and simplify.

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Step 5.4.1

Divide each term in

$- 6 x = - \frac{20}{3}$ by

$- 6$ .

$\frac{- 6 x}{- 6} = \frac{- \frac{20}{3}}{- 6}$

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.

$\frac{- 6 x}{- 6} = \frac{- \frac{20}{3}}{- 6}$

Step 5.4.2.1.2

Divide

$x$ by

$1$ .

$x = \frac{- \frac{20}{3}}{- 6}$

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.

$x = - \frac{20}{3} \cdot \frac{1}{- 6}$

Step 5.4.3.2

Cancel the common factor of

$2$ .

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Step 5.4.3.2.1

Move the leading negative in

$- \frac{20}{3}$ into the numerator.

$x = \frac{- 20}{3} \cdot \frac{1}{- 6}$

Step 5.4.3.2.2

Factor

$2$ out of

$- 20$ .

$x = \frac{2 (- 10)}{3} \cdot \frac{1}{- 6}$

Step 5.4.3.2.3

Factor

$2$ out of

$- 6$ .

$x = \frac{2 \cdot - 10}{3} \cdot \frac{1}{2 \cdot - 3}$

Step 5.4.3.2.4

Cancel the common factor.

$x = \frac{2 \cdot - 10}{3} \cdot \frac{1}{2 \cdot - 3}$

Step 5.4.3.2.5

Rewrite the expression.

$x = \frac{- 10}{3} \cdot \frac{1}{- 3}$

Step 5.4.3.3

Multiply

$\frac{- 10}{3}$ by

$\frac{1}{- 3}$ .

$x = \frac{- 10}{3 \cdot - 3}$

Step 5.4.3.4

Multiply

$3$ by

$- 3$ .

$x = \frac{- 10}{- 9}$

Step 5.4.3.5

Dividing two negative values results in a positive value.

$x = \frac{10}{9}$

Step 6

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

$(\frac{10}{9}, \frac{2}{3})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(\frac{10}{9}, \frac{2}{3})$

Equation Form:

$x = \frac{10}{9}, y = \frac{2}{3}$

Step 8

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