Algebra Examples

$\frac{4}{5} x + \frac{2}{5} y = \frac{2}{15}$ $\frac{1}{3} x - 2 y = \frac{3}{2}$

Step 1

Multiply each term in $\frac{4}{5} x + \frac{2}{5} y = \frac{2}{15}$ by $15$ to eliminate the fractions.

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Step 1.1

Multiply each term in $\frac{4}{5} x + \frac{2}{5} y = \frac{2}{15}$ by $15$ .

$\frac{4}{5} x \cdot 15 + \frac{2}{5} y \cdot 15 = \frac{2}{15} \cdot 15$

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

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

Combine $\frac{4}{5}$ and $x$ .

$\frac{4 x}{5} \cdot 15 + \frac{2}{5} y \cdot 15 = \frac{2}{15} \cdot 15$

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

Step 1.2.1.2

Cancel the common factor of $5$ .

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Step 1.2.1.2.1

Factor $5$ out of $15$ .

$\frac{4 x}{5} \cdot (5 (3)) + \frac{2}{5} y \cdot 15 = \frac{2}{15} \cdot 15$

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

Step 1.2.1.2.2

Cancel the common factor.

$\frac{4 x}{5} \cdot (5 \cdot 3) + \frac{2}{5} y \cdot 15 = \frac{2}{15} \cdot 15$

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

Step 1.2.1.2.3

Rewrite the expression.

$4 x \cdot 3 + \frac{2}{5} y \cdot 15 = \frac{2}{15} \cdot 15$

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

$4 x \cdot 3 + \frac{2}{5} y \cdot 15 = \frac{2}{15} \cdot 15$

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

Step 1.2.1.3

Multiply $3$ by $4$ .

$12 x + \frac{2}{5} y \cdot 15 = \frac{2}{15} \cdot 15$

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

Step 1.2.1.4

Combine $\frac{2}{5}$ and $y$ .

$12 x + \frac{2 y}{5} \cdot 15 = \frac{2}{15} \cdot 15$

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

Step 1.2.1.5

Cancel the common factor of $5$ .

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Step 1.2.1.5.1

Factor $5$ out of $15$ .

$12 x + \frac{2 y}{5} \cdot (5 (3)) = \frac{2}{15} \cdot 15$

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

Step 1.2.1.5.2

Cancel the common factor.

$12 x + \frac{2 y}{5} \cdot (5 \cdot 3) = \frac{2}{15} \cdot 15$

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

Step 1.2.1.5.3

Rewrite the expression.

$12 x + 2 y \cdot 3 = \frac{2}{15} \cdot 15$

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

$12 x + 2 y \cdot 3 = \frac{2}{15} \cdot 15$

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

Step 1.2.1.6

Multiply $3$ by $2$ .

$12 x + 6 y = \frac{2}{15} \cdot 15$

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

$12 x + 6 y = \frac{2}{15} \cdot 15$

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

$12 x + 6 y = \frac{2}{15} \cdot 15$

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $15$ .

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Step 1.3.1.1

Cancel the common factor.

$12 x + 6 y = \frac{2}{15} \cdot 15$

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

Step 1.3.1.2

Rewrite the expression.

$12 x + 6 y = 2$

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

$12 x + 6 y = 2$

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

$12 x + 6 y = 2$

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

$12 x + 6 y = 2$

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

Step 2

Multiply each term in $\frac{1}{3} x - 2 y = \frac{3}{2}$ by $6$ to eliminate the fractions.

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Step 2.1

Multiply each term in $\frac{1}{3} x - 2 y = \frac{3}{2}$ by $6$ .

$\frac{1}{3} x \cdot 6 - 2 y \cdot 6 = \frac{3}{2} \cdot 6$

$12 x + 6 y = 2$

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

Combine $\frac{1}{3}$ and $x$ .

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

$12 x + 6 y = 2$

Step 2.2.1.2

Cancel the common factor of $3$ .

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Step 2.2.1.2.1

Factor $3$ out of $6$ .

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

$12 x + 6 y = 2$

Step 2.2.1.2.2

Cancel the common factor.

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

$12 x + 6 y = 2$

Step 2.2.1.2.3

Rewrite the expression.

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

$12 x + 6 y = 2$

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

$12 x + 6 y = 2$

Step 2.2.1.3

Move $2$ to the left of $x$ .

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

$12 x + 6 y = 2$

Step 2.2.1.4

Multiply $6$ by $- 2$ .

$2 x - 12 y = \frac{3}{2} \cdot 6$

$12 x + 6 y = 2$

$2 x - 12 y = \frac{3}{2} \cdot 6$

$12 x + 6 y = 2$

$2 x - 12 y = \frac{3}{2} \cdot 6$

$12 x + 6 y = 2$

Step 2.3

Simplify the right side.

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Step 2.3.1

Cancel the common factor of $2$ .

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Step 2.3.1.1

Factor $2$ out of $6$ .

$2 x - 12 y = \frac{3}{2} \cdot (2 (3))$

$12 x + 6 y = 2$

Step 2.3.1.2

Cancel the common factor.

$2 x - 12 y = \frac{3}{2} \cdot (2 \cdot 3)$

$12 x + 6 y = 2$

Step 2.3.1.3

Rewrite the expression.

$2 x - 12 y = 3 \cdot 3$

$12 x + 6 y = 2$

$2 x - 12 y = 3 \cdot 3$

$12 x + 6 y = 2$

Step 2.3.2

Multiply $3$ by $3$ .

$2 x - 12 y = 9$

$12 x + 6 y = 2$

$2 x - 12 y = 9$

$12 x + 6 y = 2$

$2 x - 12 y = 9$

$12 x + 6 y = 2$

Step 3

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

$12 x + 6 y = 2$

$(- 6) \cdot (2 x - 12 y) = (- 6) (9)$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $(- 6) \cdot (2 x - 12 y)$ .

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Step 4.1.1.1

Apply the distributive property.

$12 x + 6 y = 2$

$- 6 (2 x) - 6 (- 12 y) = (- 6) (9)$

Step 4.1.1.2

Multiply.

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Step 4.1.1.2.1

Multiply $2$ by $- 6$ .

$12 x + 6 y = 2$

$- 12 x - 6 (- 12 y) = (- 6) (9)$

Step 4.1.1.2.2

Multiply $- 12$ by $- 6$ .

$12 x + 6 y = 2$

$- 12 x + 72 y = (- 6) (9)$

$12 x + 6 y = 2$

$- 12 x + 72 y = (- 6) (9)$

$12 x + 6 y = 2$

$- 12 x + 72 y = (- 6) (9)$

$12 x + 6 y = 2$

$- 12 x + 72 y = (- 6) (9)$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 6$ by $9$ .

$12 x + 6 y = 2$

$- 12 x + 72 y = - 54$

$12 x + 6 y = 2$

$- 12 x + 72 y = - 54$

$12 x + 6 y = 2$

$- 12 x + 72 y = - 54$

Step 5

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

		$1$	$2$	$x$	$+$		$6$	$y$	$=$			$2$
$+$	$-$	$1$	$2$	$x$	$+$	$7$	$2$	$y$	$=$	$-$	$5$	$4$
						$7$	$8$	$y$	$=$	$-$	$5$	$2$

Step 6

Divide each term in $78 y = - 52$ by $78$ and simplify.

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Step 6.1

Divide each term in $78 y = - 52$ by $78$ .

$\frac{78 y}{78} = \frac{- 52}{78}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $78$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{78 y}{78} = \frac{- 52}{78}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 52}{78}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Cancel the common factor of $- 52$ and $78$ .

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Step 6.3.1.1

Factor $26$ out of $- 52$ .

$y = \frac{26 (- 2)}{78}$

Step 6.3.1.2

Cancel the common factors.

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Step 6.3.1.2.1

Factor $26$ out of $78$ .

$y = \frac{26 \cdot - 2}{26 \cdot 3}$

Step 6.3.1.2.2

Cancel the common factor.

$y = \frac{26 \cdot - 2}{26 \cdot 3}$

Step 6.3.1.2.3

Rewrite the expression.

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

Step 6.3.2

Move the negative in front of the fraction.

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

Step 7

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

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Step 7.1

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

$12 x + 6 (- \frac{2}{3}) = 2$

Step 7.2

Simplify each term.

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Step 7.2.1

Cancel the common factor of $3$ .

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Step 7.2.1.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$12 x + 6 (\frac{- 2}{3}) = 2$

Step 7.2.1.2

Factor $3$ out of $6$ .

$12 x + 3 (2) \frac{- 2}{3} = 2$

Step 7.2.1.3

Cancel the common factor.

$12 x + 3 \cdot 2 \frac{- 2}{3} = 2$

Step 7.2.1.4

Rewrite the expression.

$12 x + 2 \cdot - 2 = 2$

Step 7.2.2

Multiply $2$ by $- 2$ .

$12 x - 4 = 2$

Step 7.3

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

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Step 7.3.1

Add $4$ to both sides of the equation.

$12 x = 2 + 4$

Step 7.3.2

Add $2$ and $4$ .

$12 x = 6$

Step 7.4

Divide each term in $12 x = 6$ by $12$ and simplify.

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Step 7.4.1

Divide each term in $12 x = 6$ by $12$ .

$\frac{12 x}{12} = \frac{6}{12}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of $12$ .

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Step 7.4.2.1.1

Cancel the common factor.

$\frac{12 x}{12} = \frac{6}{12}$

Step 7.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{6}{12}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Cancel the common factor of $6$ and $12$ .

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Step 7.4.3.1.1

Factor $6$ out of $6$ .

$x = \frac{6 (1)}{12}$

Step 7.4.3.1.2

Cancel the common factors.

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Step 7.4.3.1.2.1

Factor $6$ out of $12$ .

$x = \frac{6 \cdot 1}{6 \cdot 2}$

Step 7.4.3.1.2.2

Cancel the common factor.

$x = \frac{6 \cdot 1}{6 \cdot 2}$

Step 7.4.3.1.2.3

Rewrite the expression.

$x = \frac{1}{2}$

Step 8

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

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

Step 9

The result can be shown in multiple forms.

Point Form:

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

Equation Form:

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

Step 10

		$1$	$2$	$x$	$+$		$6$	$y$	$=$			$2$
$+$	$-$	$1$	$2$	$x$	$+$	$7$	$2$	$y$	$=$	$-$	$5$	$4$
						$7$	$8$	$y$	$=$	$-$	$5$	$2$

		$1$	$2$	$x$	$+$		$6$	$y$	$=$			$2$
$+$	$-$	$1$	$2$	$x$	$+$	$7$	$2$	$y$	$=$	$-$	$5$	$4$
						$7$	$8$	$y$	$=$	$-$	$5$	$2$

		$1$	$2$	$x$	$+$		$6$	$y$	$=$			$2$
$+$	$-$	$1$	$2$	$x$	$+$	$7$	$2$	$y$	$=$	$-$	$5$	$4$
						$7$	$8$	$y$	$=$	$-$	$5$	$2$