Solve by Addition/Elimination 5x+3y=6 , 3x-y=5

$5 x + 3 y = 6$ , $3 x - y = 5$

Step 1

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

$5 x + 3 y = 6$

$(3) \cdot (3 x - y) = (3) (5)$

Step 2

Simplify.

Simplify the left side.

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

Apply the distributive property.

$5 x + 3 y = 6$

$3 (3 x) + 3 (- y) = (3) (5)$

Multiply.

Multiply $3$ by $3$ .

$5 x + 3 y = 6$

$9 x + 3 (- y) = (3) (5)$

Multiply $- 1$ by $3$ .

$5 x + 3 y = 6$

$9 x - 3 y = (3) (5)$

$5 x + 3 y = 6$

$9 x - 3 y = (3) (5)$

$5 x + 3 y = 6$

$9 x - 3 y = (3) (5)$

$5 x + 3 y = 6$

$9 x - 3 y = (3) (5)$

Simplify the right side.

Multiply $3$ by $5$ .

$5 x + 3 y = 6$

$9 x - 3 y = 15$

$5 x + 3 y = 6$

$9 x - 3 y = 15$

$5 x + 3 y = 6$

$9 x - 3 y = 15$

Step 3

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

		$5$	$x$	$+$	$3$	$y$	$=$		$6$
$+$		$9$	$x$	$-$	$3$	$y$	$=$	$1$	$5$
	$1$	$4$	$x$				$=$	$2$	$1$

Step 4

Divide each term in $14 x = 21$ by $14$ and simplify.

Divide each term in $14 x = 21$ by $14$ .

$\frac{14 x}{14} = \frac{21}{14}$

Simplify the left side.

Cancel the common factor of $14$ .

Cancel the common factor.

$\frac{14 x}{14} = \frac{21}{14}$

Divide $x$ by $1$ .

$x = \frac{21}{14}$

Simplify the right side.

Cancel the common factor of $21$ and $14$ .

Factor $7$ out of $21$ .

$x = \frac{7 (3)}{14}$

Cancel the common factors.

Factor $7$ out of $14$ .

$x = \frac{7 \cdot 3}{7 \cdot 2}$

Cancel the common factor.

$x = \frac{7 \cdot 3}{7 \cdot 2}$

Rewrite the expression.

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

Step 5

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 $y$ .

$5 (\frac{3}{2}) + 3 y = 6$

Multiply $5 (\frac{3}{2})$ .

Combine $5$ and $\frac{3}{2}$ .

$\frac{5 \cdot 3}{2} + 3 y = 6$

Multiply $5$ by $3$ .

$\frac{15}{2} + 3 y = 6$

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

Subtract $\frac{15}{2}$ from both sides of the equation.

$3 y = 6 - \frac{15}{2}$

To write $6$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$3 y = 6 \cdot \frac{2}{2} - \frac{15}{2}$

Combine $6$ and $\frac{2}{2}$ .

$3 y = \frac{6 \cdot 2}{2} - \frac{15}{2}$

Combine the numerators over the common denominator.

$3 y = \frac{6 \cdot 2 - 15}{2}$

Simplify the numerator.

Multiply $6$ by $2$ .

$3 y = \frac{12 - 15}{2}$

Subtract $15$ from $12$ .

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

Move the negative in front of the fraction.

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

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

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

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

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

Cancel the common factor of $3$ .

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

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

Factor $3$ out of $- 3$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

The result can be shown in multiple forms.

Point Form:

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

Equation Form:

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

Step 8

Solve by Addition/Elimination 5x+3y=6 , 3x-y=5

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