Algebra Examples

$2 x + \frac{4}{3} y = 1$ $y - \frac{9}{13} x = 9$

Step 1

Multiply each term in $2 x + \frac{4}{3} y = 1$ by $3$ to eliminate the fractions.

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Step 1.1

Multiply each term in $2 x + \frac{4}{3} y = 1$ by $3$ .

$2 x \cdot 3 + \frac{4}{3} y \cdot 3 = 1 \cdot 3$

$y - \frac{9}{13} x = 9$

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 $2$ .

$6 x + \frac{4}{3} y \cdot 3 = 1 \cdot 3$

$y - \frac{9}{13} x = 9$

Step 1.2.1.2

Combine $\frac{4}{3}$ and $y$ .

$6 x + \frac{4 y}{3} \cdot 3 = 1 \cdot 3$

$y - \frac{9}{13} x = 9$

Step 1.2.1.3

Cancel the common factor of $3$ .

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Step 1.2.1.3.1

Cancel the common factor.

$6 x + \frac{4 y}{3} \cdot 3 = 1 \cdot 3$

$y - \frac{9}{13} x = 9$

Step 1.2.1.3.2

Rewrite the expression.

$6 x + 4 y = 1 \cdot 3$

$y - \frac{9}{13} x = 9$

$6 x + 4 y = 1 \cdot 3$

$y - \frac{9}{13} x = 9$

$6 x + 4 y = 1 \cdot 3$

$y - \frac{9}{13} x = 9$

$6 x + 4 y = 1 \cdot 3$

$y - \frac{9}{13} x = 9$

Step 1.3

Simplify the right side.

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Step 1.3.1

Multiply $3$ by $1$ .

$6 x + 4 y = 3$

$y - \frac{9}{13} x = 9$

$6 x + 4 y = 3$

$y - \frac{9}{13} x = 9$

$6 x + 4 y = 3$

$y - \frac{9}{13} x = 9$

Step 2

Multiply each term in $y - \frac{9}{13} x = 9$ by $13$ to eliminate the fractions.

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Step 2.1

Multiply each term in $y - \frac{9}{13} x = 9$ by $13$ .

$y \cdot 13 - \frac{9}{13} x \cdot 13 = 9 \cdot 13$

$6 x + 4 y = 3$

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

Move $13$ to the left of $y$ .

$13 \cdot y - \frac{9}{13} x \cdot 13 = 9 \cdot 13$

$6 x + 4 y = 3$

Step 2.2.1.2

Combine $x$ and $\frac{9}{13}$ .

$13 y - \frac{x \cdot 9}{13} \cdot 13 = 9 \cdot 13$

$6 x + 4 y = 3$

Step 2.2.1.3

Cancel the common factor of $13$ .

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Step 2.2.1.3.1

Move the leading negative in $- \frac{x \cdot 9}{13}$ into the numerator.

$13 y + \frac{- x \cdot 9}{13} \cdot 13 = 9 \cdot 13$

$6 x + 4 y = 3$

Step 2.2.1.3.2

Cancel the common factor.

$13 y + \frac{- x \cdot 9}{13} \cdot 13 = 9 \cdot 13$

$6 x + 4 y = 3$

Step 2.2.1.3.3

Rewrite the expression.

$13 y - x \cdot 9 = 9 \cdot 13$

$6 x + 4 y = 3$

$13 y - x \cdot 9 = 9 \cdot 13$

$6 x + 4 y = 3$

Step 2.2.1.4

Multiply $9$ by $- 1$ .

$13 y - 9 x = 9 \cdot 13$

$6 x + 4 y = 3$

$13 y - 9 x = 9 \cdot 13$

$6 x + 4 y = 3$

$13 y - 9 x = 9 \cdot 13$

$6 x + 4 y = 3$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $9$ by $13$ .

$13 y - 9 x = 117$

$6 x + 4 y = 3$

$13 y - 9 x = 117$

$6 x + 4 y = 3$

$13 y - 9 x = 117$

$6 x + 4 y = 3$

Step 3

Simplify the left side.

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Step 3.1

Reorder $13 y$ and $- 9 x$ .

$- 9 x + 13 y = 117$

$6 x + 4 y = 3$

$- 9 x + 13 y = 117$

$6 x + 4 y = 3$

Step 4

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

$(3) \cdot (6 x + 4 y) = (3) (3)$

$(2) \cdot (- 9 x + 13 y) = (2) (117)$

Step 5

Simplify.

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Step 5.1

Simplify the left side.

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Step 5.1.1

Simplify $(3) \cdot (6 x + 4 y)$ .

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Step 5.1.1.1

Apply the distributive property.

$3 (6 x) + 3 (4 y) = (3) (3)$

$(2) \cdot (- 9 x + 13 y) = (2) (117)$

Step 5.1.1.2

Multiply.

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Step 5.1.1.2.1

Multiply $6$ by $3$ .

$18 x + 3 (4 y) = (3) (3)$

$(2) \cdot (- 9 x + 13 y) = (2) (117)$

Step 5.1.1.2.2

Multiply $4$ by $3$ .

$18 x + 12 y = (3) (3)$

$(2) \cdot (- 9 x + 13 y) = (2) (117)$

$18 x + 12 y = (3) (3)$

$(2) \cdot (- 9 x + 13 y) = (2) (117)$

$18 x + 12 y = (3) (3)$

$(2) \cdot (- 9 x + 13 y) = (2) (117)$

$18 x + 12 y = (3) (3)$

$(2) \cdot (- 9 x + 13 y) = (2) (117)$

Step 5.2

Simplify the right side.

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Step 5.2.1

Multiply $3$ by $3$ .

$18 x + 12 y = 9$

$(2) \cdot (- 9 x + 13 y) = (2) (117)$

$18 x + 12 y = 9$

$(2) \cdot (- 9 x + 13 y) = (2) (117)$

Step 5.3

Simplify the left side.

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Step 5.3.1

Simplify $(2) \cdot (- 9 x + 13 y)$ .

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Step 5.3.1.1

Apply the distributive property.

$18 x + 12 y = 9$

$2 (- 9 x) + 2 (13 y) = (2) (117)$

Step 5.3.1.2

Multiply.

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Step 5.3.1.2.1

Multiply $- 9$ by $2$ .

$18 x + 12 y = 9$

$- 18 x + 2 (13 y) = (2) (117)$

Step 5.3.1.2.2

Multiply $13$ by $2$ .

$18 x + 12 y = 9$

$- 18 x + 26 y = (2) (117)$

$18 x + 12 y = 9$

$- 18 x + 26 y = (2) (117)$

$18 x + 12 y = 9$

$- 18 x + 26 y = (2) (117)$

$18 x + 12 y = 9$

$- 18 x + 26 y = (2) (117)$

Step 5.4

Simplify the right side.

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Step 5.4.1

Multiply $2$ by $117$ .

$18 x + 12 y = 9$

$- 18 x + 26 y = 234$

$18 x + 12 y = 9$

$- 18 x + 26 y = 234$

$18 x + 12 y = 9$

$- 18 x + 26 y = 234$

Step 6

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

		$1$	$8$	$x$	$+$	$1$	$2$	$y$	$=$			$9$
$+$	$-$	$1$	$8$	$x$	$+$	$2$	$6$	$y$	$=$	$2$	$3$	$4$
						$3$	$8$	$y$	$=$	$2$	$4$	$3$

Step 7

Divide each term in $38 y = 243$ by $38$ and simplify.

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Step 7.1

Divide each term in $38 y = 243$ by $38$ .

$\frac{38 y}{38} = \frac{243}{38}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $38$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{38 y}{38} = \frac{243}{38}$

Step 7.2.1.2

Divide $y$ by $1$ .

$y = \frac{243}{38}$

Step 8

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

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Step 8.1

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

$18 x + 12 (\frac{243}{38}) = 9$

Step 8.2

Simplify each term.

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Step 8.2.1

Cancel the common factor of $2$ .

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Step 8.2.1.1

Factor $2$ out of $12$ .

$18 x + 2 (6) \frac{243}{38} = 9$

Step 8.2.1.2

Factor $2$ out of $38$ .

$18 x + 2 \cdot 6 \frac{243}{2 \cdot 19} = 9$

Step 8.2.1.3

Cancel the common factor.

$18 x + 2 \cdot 6 \frac{243}{2 \cdot 19} = 9$

Step 8.2.1.4

Rewrite the expression.

$18 x + 6 (\frac{243}{19}) = 9$

Step 8.2.2

Combine $6$ and $\frac{243}{19}$ .

$18 x + \frac{6 \cdot 243}{19} = 9$

Step 8.2.3

Multiply $6$ by $243$ .

$18 x + \frac{1458}{19} = 9$

Step 8.3

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

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Step 8.3.1

Subtract $\frac{1458}{19}$ from both sides of the equation.

$18 x = 9 - \frac{1458}{19}$

Step 8.3.2

To write $9$ as a fraction with a common denominator, multiply by $\frac{19}{19}$ .

$18 x = 9 \cdot \frac{19}{19} - \frac{1458}{19}$

Step 8.3.3

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

$18 x = \frac{9 \cdot 19}{19} - \frac{1458}{19}$

Step 8.3.4

Combine the numerators over the common denominator.

$18 x = \frac{9 \cdot 19 - 1458}{19}$

Step 8.3.5

Simplify the numerator.

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Step 8.3.5.1

Multiply $9$ by $19$ .

$18 x = \frac{171 - 1458}{19}$

Step 8.3.5.2

Subtract $1458$ from $171$ .

$18 x = \frac{- 1287}{19}$

Step 8.3.6

Move the negative in front of the fraction.

$18 x = - \frac{1287}{19}$

Step 8.4

Divide each term in $18 x = - \frac{1287}{19}$ by $18$ and simplify.

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Step 8.4.1

Divide each term in $18 x = - \frac{1287}{19}$ by $18$ .

$\frac{18 x}{18} = \frac{- \frac{1287}{19}}{18}$

Step 8.4.2

Simplify the left side.

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Step 8.4.2.1

Cancel the common factor of $18$ .

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Step 8.4.2.1.1

Cancel the common factor.

$\frac{18 x}{18} = \frac{- \frac{1287}{19}}{18}$

Step 8.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- \frac{1287}{19}}{18}$

Step 8.4.3

Simplify the right side.

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Step 8.4.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{1287}{19} \cdot \frac{1}{18}$

Step 8.4.3.2

Cancel the common factor of $9$ .

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Step 8.4.3.2.1

Move the leading negative in $- \frac{1287}{19}$ into the numerator.

$x = \frac{- 1287}{19} \cdot \frac{1}{18}$

Step 8.4.3.2.2

Factor $9$ out of $- 1287$ .

$x = \frac{9 (- 143)}{19} \cdot \frac{1}{18}$

Step 8.4.3.2.3

Factor $9$ out of $18$ .

$x = \frac{9 \cdot - 143}{19} \cdot \frac{1}{9 \cdot 2}$

Step 8.4.3.2.4

Cancel the common factor.

$x = \frac{9 \cdot - 143}{19} \cdot \frac{1}{9 \cdot 2}$

Step 8.4.3.2.5

Rewrite the expression.

$x = \frac{- 143}{19} \cdot \frac{1}{2}$

Step 8.4.3.3

Multiply $\frac{- 143}{19}$ by $\frac{1}{2}$ .

$x = \frac{- 143}{19 \cdot 2}$

Step 8.4.3.4

Simplify the expression.

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Step 8.4.3.4.1

Multiply $19$ by $2$ .

$x = \frac{- 143}{38}$

Step 8.4.3.4.2

Move the negative in front of the fraction.

$x = - \frac{143}{38}$

Step 9

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

$(- \frac{143}{38}, \frac{243}{38})$

Step 10

The result can be shown in multiple forms.

Point Form:

$(- \frac{143}{38}, \frac{243}{38})$

Equation Form:

$x = - \frac{143}{38}, y = \frac{243}{38}$

Step 11

		$1$	$8$	$x$	$+$	$1$	$2$	$y$	$=$			$9$
$+$	$-$	$1$	$8$	$x$	$+$	$2$	$6$	$y$	$=$	$2$	$3$	$4$
						$3$	$8$	$y$	$=$	$2$	$4$	$3$

		$1$	$8$	$x$	$+$	$1$	$2$	$y$	$=$			$9$
$+$	$-$	$1$	$8$	$x$	$+$	$2$	$6$	$y$	$=$	$2$	$3$	$4$
						$3$	$8$	$y$	$=$	$2$	$4$	$3$

		$1$	$8$	$x$	$+$	$1$	$2$	$y$	$=$			$9$
$+$	$-$	$1$	$8$	$x$	$+$	$2$	$6$	$y$	$=$	$2$	$3$	$4$
						$3$	$8$	$y$	$=$	$2$	$4$	$3$