Algebra Examples

$3 x + y = \frac{1}{3}$ $2 x - 3 y = \frac{8}{3}$

Step 1

Multiply each term in $3 x + y = \frac{1}{3}$ by $3$ to eliminate the fractions.

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Step 1.1

Multiply each term in $3 x + y = \frac{1}{3}$ by $3$ .

$3 x \cdot 3 + y \cdot 3 = \frac{1}{3} \cdot 3$

$2 x - 3 y = \frac{8}{3}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Multiply $3$ by $3$ .

$9 x + y \cdot 3 = \frac{1}{3} \cdot 3$

$2 x - 3 y = \frac{8}{3}$

Step 1.2.1.2

Move $3$ to the left of $y$ .

$9 x + 3 y = \frac{1}{3} \cdot 3$

$2 x - 3 y = \frac{8}{3}$

$9 x + 3 y = \frac{1}{3} \cdot 3$

$2 x - 3 y = \frac{8}{3}$

$9 x + 3 y = \frac{1}{3} \cdot 3$

$2 x - 3 y = \frac{8}{3}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $3$ .

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Step 1.3.1.1

Cancel the common factor.

$9 x + 3 y = \frac{1}{3} \cdot 3$

$2 x - 3 y = \frac{8}{3}$

Step 1.3.1.2

Rewrite the expression.

$9 x + 3 y = 1$

$2 x - 3 y = \frac{8}{3}$

$9 x + 3 y = 1$

$2 x - 3 y = \frac{8}{3}$

$9 x + 3 y = 1$

$2 x - 3 y = \frac{8}{3}$

$9 x + 3 y = 1$

$2 x - 3 y = \frac{8}{3}$

Step 2

Multiply each term in $2 x - 3 y = \frac{8}{3}$ by $3$ to eliminate the fractions.

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Step 2.1

Multiply each term in $2 x - 3 y = \frac{8}{3}$ by $3$ .

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

$9 x + 3 y = 1$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $3$ by $2$ .

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

$9 x + 3 y = 1$

Step 2.2.1.2

Multiply $3$ by $- 3$ .

$6 x - 9 y = \frac{8}{3} \cdot 3$

$9 x + 3 y = 1$

$6 x - 9 y = \frac{8}{3} \cdot 3$

$9 x + 3 y = 1$

$6 x - 9 y = \frac{8}{3} \cdot 3$

$9 x + 3 y = 1$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $3$ .

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Step 2.3.1.1

Cancel the common factor.

$6 x - 9 y = \frac{8}{3} \cdot 3$

$9 x + 3 y = 1$

Step 2.3.1.2

Rewrite the expression.

$6 x - 9 y = 8$

$9 x + 3 y = 1$

$6 x - 9 y = 8$

$9 x + 3 y = 1$

$6 x - 9 y = 8$

$9 x + 3 y = 1$

$6 x - 9 y = 8$

$9 x + 3 y = 1$

Step 3

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

$(3) \cdot (9 x + 3 y) = (3) (1)$

$6 x - 9 y = 8$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $(3) \cdot (9 x + 3 y)$ .

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Step 4.1.1.1

Apply the distributive property.

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

$6 x - 9 y = 8$

Step 4.1.1.2

Multiply.

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Step 4.1.1.2.1

Multiply $9$ by $3$ .

$27 x + 3 (3 y) = (3) (1)$

$6 x - 9 y = 8$

Step 4.1.1.2.2

Multiply $3$ by $3$ .

$27 x + 9 y = (3) (1)$

$6 x - 9 y = 8$

$27 x + 9 y = (3) (1)$

$6 x - 9 y = 8$

$27 x + 9 y = (3) (1)$

$6 x - 9 y = 8$

$27 x + 9 y = (3) (1)$

$6 x - 9 y = 8$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $3$ by $1$ .

$27 x + 9 y = 3$

$6 x - 9 y = 8$

$27 x + 9 y = 3$

$6 x - 9 y = 8$

$27 x + 9 y = 3$

$6 x - 9 y = 8$

Step 5

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

	$2$	$7$	$x$	$+$	$9$	$y$	$=$	$3$
$+$		$6$	$x$	$-$	$9$	$y$	$=$	$8$
	$3$	$3$	$x$				$=$	$1$	$1$

Step 6

Divide each term in $33 x = 11$ by $33$ and simplify.

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Step 6.1

Divide each term in $33 x = 11$ by $33$ .

$\frac{33 x}{33} = \frac{11}{33}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $33$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{33 x}{33} = \frac{11}{33}$

Step 6.2.1.2

Divide $x$ by $1$ .

$x = \frac{11}{33}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $11$ and $33$ .

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Step 6.3.1.1

Factor $11$ out of $11$ .

$x = \frac{11 (1)}{33}$

Step 6.3.1.2

Cancel the common factors.

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Step 6.3.1.2.1

Factor $11$ out of $33$ .

$x = \frac{11 \cdot 1}{11 \cdot 3}$

Step 6.3.1.2.2

Cancel the common factor.

$x = \frac{11 \cdot 1}{11 \cdot 3}$

Step 6.3.1.2.3

Rewrite the expression.

$x = \frac{1}{3}$

Step 7

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

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Step 7.1

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

$27 (\frac{1}{3}) + 9 y = 3$

Step 7.2

Cancel the common factor of $3$ .

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Step 7.2.1

Factor $3$ out of $27$ .

$3 (9) \frac{1}{3} + 9 y = 3$

Step 7.2.2

Cancel the common factor.

$3 \cdot 9 \frac{1}{3} + 9 y = 3$

Step 7.2.3

Rewrite the expression.

$9 + 9 y = 3$

Step 7.3

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

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Step 7.3.1

Subtract $9$ from both sides of the equation.

$9 y = 3 - 9$

Step 7.3.2

Subtract $9$ from $3$ .

$9 y = - 6$

Step 7.4

Divide each term in $9 y = - 6$ by $9$ and simplify.

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Step 7.4.1

Divide each term in $9 y = - 6$ by $9$ .

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

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of $9$ .

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Step 7.4.2.1.1

Cancel the common factor.

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

Step 7.4.2.1.2

Divide $y$ by $1$ .

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

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Cancel the common factor of $- 6$ and $9$ .

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Step 7.4.3.1.1

Factor $3$ out of $- 6$ .

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

Step 7.4.3.1.2

Cancel the common factors.

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Step 7.4.3.1.2.1

Factor $3$ out of $9$ .

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

Step 7.4.3.1.2.2

Cancel the common factor.

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

Step 7.4.3.1.2.3

Rewrite the expression.

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

Step 7.4.3.2

Move the negative in front of the fraction.

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

Step 8

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

$(\frac{1}{3}, - \frac{2}{3})$

Step 9

The result can be shown in multiple forms.

Point Form:

$(\frac{1}{3}, - \frac{2}{3})$

Equation Form:

$x = \frac{1}{3}, y = - \frac{2}{3}$

Step 10

	$2$	$7$	$x$	$+$	$9$	$y$	$=$	$3$
$+$		$6$	$x$	$-$	$9$	$y$	$=$	$8$
	$3$	$3$	$x$				$=$	$1$	$1$

	$2$	$7$	$x$	$+$	$9$	$y$	$=$	$3$
$+$		$6$	$x$	$-$	$9$	$y$	$=$	$8$
	$3$	$3$	$x$				$=$	$1$	$1$

	$2$	$7$	$x$	$+$	$9$	$y$	$=$	$3$
$+$		$6$	$x$	$-$	$9$	$y$	$=$	$8$
	$3$	$3$	$x$				$=$	$1$	$1$