Solve by Addition/Elimination 2x+3y=6 , x-3y=7

$2 x + 3 y = 6$ , $x - 3 y = 7$

Step 1

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

$2 x + 3 y = 6$

$(- 2) \cdot (x - 3 y) = (- 2) (7)$

Step 2

Simplify.

Simplify the left side.

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

Apply the distributive property.

$2 x + 3 y = 6$

$- 2 x - 2 (- 3 y) = (- 2) (7)$

Multiply $- 3$ by $- 2$ .

$2 x + 3 y = 6$

$- 2 x + 6 y = (- 2) (7)$

$2 x + 3 y = 6$

$- 2 x + 6 y = (- 2) (7)$

$2 x + 3 y = 6$

$- 2 x + 6 y = (- 2) (7)$

Simplify the right side.

Multiply $- 2$ by $7$ .

$2 x + 3 y = 6$

$- 2 x + 6 y = - 14$

$2 x + 3 y = 6$

$- 2 x + 6 y = - 14$

$2 x + 3 y = 6$

$- 2 x + 6 y = - 14$

Step 3

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

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

Step 4

Divide each term in $9 y = - 8$ by $9$ and simplify.

Divide each term in $9 y = - 8$ by $9$ .

$\frac{9 y}{9} = \frac{- 8}{9}$

Simplify the left side.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{9 y}{9} = \frac{- 8}{9}$

Divide $y$ by $1$ .

$y = \frac{- 8}{9}$

Simplify the right side.

Move the negative in front of the fraction.

$y = - \frac{8}{9}$

Step 5

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

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

$2 x + 3 (- \frac{8}{9}) = 6$

Simplify each term.

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{8}{9}$ into the numerator.

$2 x + 3 (\frac{- 8}{9}) = 6$

Factor $3$ out of $9$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

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

Add $\frac{8}{3}$ to both sides of the equation.

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

To write $6$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Combine $6$ and $\frac{3}{3}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $6$ by $3$ .

$2 x = \frac{18 + 8}{3}$

Add $18$ and $8$ .

$2 x = \frac{26}{3}$

Divide each term in $2 x = \frac{26}{3}$ by $2$ and simplify.

Divide each term in $2 x = \frac{26}{3}$ by $2$ .

$\frac{2 x}{2} = \frac{\frac{26}{3}}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{\frac{26}{3}}{2}$

Divide $x$ by $1$ .

$x = \frac{\frac{26}{3}}{2}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{26}{3} \cdot \frac{1}{2}$

Cancel the common factor of $2$ .

Factor $2$ out of $26$ .

$x = \frac{2 (13)}{3} \cdot \frac{1}{2}$

Cancel the common factor.

$x = \frac{2 \cdot 13}{3} \cdot \frac{1}{2}$

Rewrite the expression.

$x = \frac{13}{3}$

Step 6

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

$(\frac{13}{3}, - \frac{8}{9})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(\frac{13}{3}, - \frac{8}{9})$

Equation Form:

$x = \frac{13}{3}, y = - \frac{8}{9}$

Step 8

Solve by Addition/Elimination 2x+3y=6 , x-3y=7

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