Solve by Addition/Elimination -x=-3y+16 5x+3y=19

$- x = - 3 y + 16$ $5 x + 3 y = 19$

Step 1

Add $3 y$ to both sides of the equation.

$- x + 3 y = 16, 5 x + 3 y = 19$

Step 2

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

$(- 1) \cdot (- x + 3 y) = (- 1) (16)$

$5 x + 3 y = 19$

Step 3

Simplify.

Simplify the left side.

Simplify $(- 1) \cdot (- x + 3 y)$ .

Apply the distributive property.

$- 1 (- x) - 1 (3 y) = (- 1) (16)$

$5 x + 3 y = 19$

Multiply $- 1 (- x)$ .

Multiply $- 1$ by $- 1$ .

$1 x - 1 (3 y) = (- 1) (16)$

$5 x + 3 y = 19$

Multiply $x$ by $1$ .

$x - 1 (3 y) = (- 1) (16)$

$5 x + 3 y = 19$

$x - 1 (3 y) = (- 1) (16)$

$5 x + 3 y = 19$

Multiply $3$ by $- 1$ .

$x - 3 y = (- 1) (16)$

$5 x + 3 y = 19$

$x - 3 y = (- 1) (16)$

$5 x + 3 y = 19$

$x - 3 y = (- 1) (16)$

$5 x + 3 y = 19$

Simplify the right side.

Multiply $- 1$ by $16$ .

$x - 3 y = - 16$

$5 x + 3 y = 19$

$x - 3 y = - 16$

$5 x + 3 y = 19$

$x - 3 y = - 16$

$5 x + 3 y = 19$

Step 4

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

		$x$	$-$	$3$	$y$	$=$	$-$	$1$	$6$
$+$	$5$	$x$	$+$	$3$	$y$	$=$		$1$	$9$
	$6$	$x$				$=$			$3$

Step 5

Divide each term in $6 x = 3$ by $6$ and simplify.

Divide each term in $6 x = 3$ by $6$ .

$\frac{6 x}{6} = \frac{3}{6}$

Simplify the left side.

Cancel the common factor of $6$ .

Cancel the common factor.

$\frac{6 x}{6} = \frac{3}{6}$

Divide $x$ by $1$ .

$x = \frac{3}{6}$

Simplify the right side.

Cancel the common factor of $3$ and $6$ .

Factor $3$ out of $3$ .

$x = \frac{3 (1)}{6}$

Cancel the common factors.

Factor $3$ out of $6$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 6

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

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

$(\frac{1}{2}) - 3 y = - 16$

Remove parentheses.

$\frac{1}{2} - 3 y = - 16$

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

Subtract $\frac{1}{2}$ from both sides of the equation.

$- 3 y = - 16 - \frac{1}{2}$

To write $- 16$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 3 y = - 16 \cdot \frac{2}{2} - \frac{1}{2}$

Combine $- 16$ and $\frac{2}{2}$ .

$- 3 y = \frac{- 16 \cdot 2}{2} - \frac{1}{2}$

Combine the numerators over the common denominator.

$- 3 y = \frac{- 16 \cdot 2 - 1}{2}$

Simplify the numerator.

Multiply $- 16$ by $2$ .

$- 3 y = \frac{- 32 - 1}{2}$

Subtract $1$ from $- 32$ .

$- 3 y = \frac{- 33}{2}$

Move the negative in front of the fraction.

$- 3 y = - \frac{33}{2}$

Divide each term in $- 3 y = - \frac{33}{2}$ by $- 3$ and simplify.

Divide each term in $- 3 y = - \frac{33}{2}$ by $- 3$ .

$\frac{- 3 y}{- 3} = \frac{- \frac{33}{2}}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 y}{- 3} = \frac{- \frac{33}{2}}{- 3}$

Divide $y$ by $1$ .

$y = \frac{- \frac{33}{2}}{- 3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$y = - \frac{33}{2} \cdot \frac{1}{- 3}$

Cancel the common factor of $3$ .

Move the leading negative in $- \frac{33}{2}$ into the numerator.

$y = \frac{- 33}{2} \cdot \frac{1}{- 3}$

Factor $3$ out of $- 33$ .

$y = \frac{3 (- 11)}{2} \cdot \frac{1}{- 3}$

Factor $3$ out of $- 3$ .

$y = \frac{3 \cdot - 11}{2} \cdot \frac{1}{3 \cdot - 1}$

Cancel the common factor.

$y = \frac{3 \cdot - 11}{2} \cdot \frac{1}{3 \cdot - 1}$

Rewrite the expression.

$y = \frac{- 11}{2} \cdot \frac{1}{- 1}$

Multiply $\frac{- 11}{2}$ by $\frac{1}{- 1}$ .

$y = \frac{- 11}{2 \cdot - 1}$

Multiply $2$ by $- 1$ .

$y = \frac{- 11}{- 2}$

Dividing two negative values results in a positive value.

$y = \frac{11}{2}$

Step 7

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

$(\frac{1}{2}, \frac{11}{2})$

Step 8

The result can be shown in multiple forms.

Point Form:

$(\frac{1}{2}, \frac{11}{2})$

Equation Form:

$x = \frac{1}{2}, y = \frac{11}{2}$

Step 9

Solve by Addition/Elimination -x=-3y+16 5x+3y=19

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