Algebra Examples

$\frac{2}{3} x + \frac{3}{4} y = \frac{7}{12}$ $\frac{1}{2} x - 2 y = 3$

Step 1

Multiply each term in $\frac{2}{3} x + \frac{3}{4} y = \frac{7}{12}$ by $12$ to eliminate the fractions.

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Step 1.1

Multiply each term in $\frac{2}{3} x + \frac{3}{4} y = \frac{7}{12}$ by $12$ .

$\frac{2}{3} x \cdot 12 + \frac{3}{4} y \cdot 12 = \frac{7}{12} \cdot 12$

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

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

Combine $\frac{2}{3}$ and $x$ .

$\frac{2 x}{3} \cdot 12 + \frac{3}{4} y \cdot 12 = \frac{7}{12} \cdot 12$

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

Step 1.2.1.2

Cancel the common factor of $3$ .

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Step 1.2.1.2.1

Factor $3$ out of $12$ .

$\frac{2 x}{3} \cdot (3 (4)) + \frac{3}{4} y \cdot 12 = \frac{7}{12} \cdot 12$

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

Step 1.2.1.2.2

Cancel the common factor.

$\frac{2 x}{3} \cdot (3 \cdot 4) + \frac{3}{4} y \cdot 12 = \frac{7}{12} \cdot 12$

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

Step 1.2.1.2.3

Rewrite the expression.

$2 x \cdot 4 + \frac{3}{4} y \cdot 12 = \frac{7}{12} \cdot 12$

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

$2 x \cdot 4 + \frac{3}{4} y \cdot 12 = \frac{7}{12} \cdot 12$

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

Step 1.2.1.3

Multiply $4$ by $2$ .

$8 x + \frac{3}{4} y \cdot 12 = \frac{7}{12} \cdot 12$

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

Step 1.2.1.4

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

$8 x + \frac{3 y}{4} \cdot 12 = \frac{7}{12} \cdot 12$

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

Step 1.2.1.5

Cancel the common factor of $4$ .

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Step 1.2.1.5.1

Factor $4$ out of $12$ .

$8 x + \frac{3 y}{4} \cdot (4 (3)) = \frac{7}{12} \cdot 12$

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

Step 1.2.1.5.2

Cancel the common factor.

$8 x + \frac{3 y}{4} \cdot (4 \cdot 3) = \frac{7}{12} \cdot 12$

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

Step 1.2.1.5.3

Rewrite the expression.

$8 x + 3 y \cdot 3 = \frac{7}{12} \cdot 12$

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

$8 x + 3 y \cdot 3 = \frac{7}{12} \cdot 12$

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

Step 1.2.1.6

Multiply $3$ by $3$ .

$8 x + 9 y = \frac{7}{12} \cdot 12$

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

$8 x + 9 y = \frac{7}{12} \cdot 12$

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

$8 x + 9 y = \frac{7}{12} \cdot 12$

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

Step 1.3

Simplify the right side.

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Step 1.3.1

Cancel the common factor of $12$ .

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Step 1.3.1.1

Cancel the common factor.

$8 x + 9 y = \frac{7}{12} \cdot 12$

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

Step 1.3.1.2

Rewrite the expression.

$8 x + 9 y = 7$

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

$8 x + 9 y = 7$

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

$8 x + 9 y = 7$

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

$8 x + 9 y = 7$

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

Step 2

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

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Step 2.1

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

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

$8 x + 9 y = 7$

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 $\frac{1}{2}$ and $x$ .

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

$8 x + 9 y = 7$

Step 2.2.1.2

Cancel the common factor of $2$ .

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Step 2.2.1.2.1

Cancel the common factor.

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

$8 x + 9 y = 7$

Step 2.2.1.2.2

Rewrite the expression.

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

$8 x + 9 y = 7$

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

$8 x + 9 y = 7$

Step 2.2.1.3

Multiply $2$ by $- 2$ .

$x - 4 y = 3 \cdot 2$

$8 x + 9 y = 7$

$x - 4 y = 3 \cdot 2$

$8 x + 9 y = 7$

$x - 4 y = 3 \cdot 2$

$8 x + 9 y = 7$

Step 2.3

Simplify the right side.

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Step 2.3.1

Multiply $3$ by $2$ .

$x - 4 y = 6$

$8 x + 9 y = 7$

$x - 4 y = 6$

$8 x + 9 y = 7$

$x - 4 y = 6$

$8 x + 9 y = 7$

Step 3

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

$8 x + 9 y = 7$

$(- 8) \cdot (x - 4 y) = (- 8) (6)$

Step 4

Simplify.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $(- 8) \cdot (x - 4 y)$ .

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Step 4.1.1.1

Apply the distributive property.

$8 x + 9 y = 7$

$- 8 x - 8 (- 4 y) = (- 8) (6)$

Step 4.1.1.2

Multiply $- 4$ by $- 8$ .

$8 x + 9 y = 7$

$- 8 x + 32 y = (- 8) (6)$

$8 x + 9 y = 7$

$- 8 x + 32 y = (- 8) (6)$

$8 x + 9 y = 7$

$- 8 x + 32 y = (- 8) (6)$

Step 4.2

Simplify the right side.

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Step 4.2.1

Multiply $- 8$ by $6$ .

$8 x + 9 y = 7$

$- 8 x + 32 y = - 48$

$8 x + 9 y = 7$

$- 8 x + 32 y = - 48$

$8 x + 9 y = 7$

$- 8 x + 32 y = - 48$

Step 5

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

		$8$	$x$	$+$		$9$	$y$	$=$			$7$
$+$	$-$	$8$	$x$	$+$	$3$	$2$	$y$	$=$	$-$	$4$	$8$
					$4$	$1$	$y$	$=$	$-$	$4$	$1$

Step 6

Divide each term in $41 y = - 41$ by $41$ and simplify.

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Step 6.1

Divide each term in $41 y = - 41$ by $41$ .

$\frac{41 y}{41} = \frac{- 41}{41}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $41$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{41 y}{41} = \frac{- 41}{41}$

Step 6.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 41}{41}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Divide $- 41$ by $41$ .

$y = - 1$

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 $x$ .

$8 x + 9 (- 1) = 7$

Step 7.2

Multiply $9$ by $- 1$ .

$8 x - 9 = 7$

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 $9$ to both sides of the equation.

$8 x = 7 + 9$

Step 7.3.2

Add $7$ and $9$ .

$8 x = 16$

Step 7.4

Divide each term in $8 x = 16$ by $8$ and simplify.

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Step 7.4.1

Divide each term in $8 x = 16$ by $8$ .

$\frac{8 x}{8} = \frac{16}{8}$

Step 7.4.2

Simplify the left side.

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Step 7.4.2.1

Cancel the common factor of $8$ .

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Step 7.4.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{16}{8}$

Step 7.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{16}{8}$

Step 7.4.3

Simplify the right side.

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Step 7.4.3.1

Divide $16$ by $8$ .

$x = 2$

Step 8

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

$(2, - 1)$

Step 9

The result can be shown in multiple forms.

Point Form:

$(2, - 1)$

Equation Form:

$x = 2, y = - 1$

Step 10

		$8$	$x$	$+$		$9$	$y$	$=$			$7$
$+$	$-$	$8$	$x$	$+$	$3$	$2$	$y$	$=$	$-$	$4$	$8$
					$4$	$1$	$y$	$=$	$-$	$4$	$1$

		$8$	$x$	$+$		$9$	$y$	$=$			$7$
$+$	$-$	$8$	$x$	$+$	$3$	$2$	$y$	$=$	$-$	$4$	$8$
					$4$	$1$	$y$	$=$	$-$	$4$	$1$

		$8$	$x$	$+$		$9$	$y$	$=$			$7$
$+$	$-$	$8$	$x$	$+$	$3$	$2$	$y$	$=$	$-$	$4$	$8$
					$4$	$1$	$y$	$=$	$-$	$4$	$1$