Solve by Addition/Elimination 3x^2-5y^2=-33 , 2x^2+3y^2=35

$3 x^{2} - 5 y^{2} = - 33$ , $2 x^{2} + 3 y^{2} = 35$

Step 1

Simplify the left side.

Reorder $3 x^{2}$ and $- 5 y^{2}$ .

$- 5 y^{2} + 3 x^{2} = - 33$

$2 x^{2} + 3 y^{2} = 35$

$- 5 y^{2} + 3 x^{2} = - 33$

$2 x^{2} + 3 y^{2} = 35$

Step 2

Simplify the left side.

Reorder $2 x^{2}$ and $3 y^{2}$ .

$3 y^{2} + 2 x^{2} = 35$

$- 5 y^{2} + 3 x^{2} = - 33$

$3 y^{2} + 2 x^{2} = 35$

$- 5 y^{2} + 3 x^{2} = - 33$

Step 3

Multiply each equation by the value that makes the coefficients of $x^{2}$ opposite.

$- 2 \cdot (- 5 y^{2} + 3 x^{2}) = (- 2) (- 33)$

$3 \cdot (3 y^{2} + 2 x^{2}) = (3) (35)$

Step 4

Simplify.

Simplify the left side.

Simplify $- 2 \cdot (- 5 y^{2} + 3 x^{2})$ .

Apply the distributive property.

$- 2 (- 5 y^{2}) - 2 (3 x^{2}) = (- 2) (- 33)$

$3 \cdot (3 y^{2} + 2 x^{2}) = (3) (35)$

Multiply.

Multiply $- 5$ by $- 2$ .

$10 y^{2} - 2 (3 x^{2}) = (- 2) (- 33)$

$3 \cdot (3 y^{2} + 2 x^{2}) = (3) (35)$

Multiply $3$ by $- 2$ .

$10 y^{2} - 6 x^{2} = (- 2) (- 33)$

$3 \cdot (3 y^{2} + 2 x^{2}) = (3) (35)$

$10 y^{2} - 6 x^{2} = (- 2) (- 33)$

$3 \cdot (3 y^{2} + 2 x^{2}) = (3) (35)$

$10 y^{2} - 6 x^{2} = (- 2) (- 33)$

$3 \cdot (3 y^{2} + 2 x^{2}) = (3) (35)$

$10 y^{2} - 6 x^{2} = (- 2) (- 33)$

$3 \cdot (3 y^{2} + 2 x^{2}) = (3) (35)$

Simplify the right side.

Multiply $- 2$ by $- 33$ .

$10 y^{2} - 6 x^{2} = 66$

$3 \cdot (3 y^{2} + 2 x^{2}) = (3) (35)$

$10 y^{2} - 6 x^{2} = 66$

$3 \cdot (3 y^{2} + 2 x^{2}) = (3) (35)$

Simplify the left side.

Simplify $3 \cdot (3 y^{2} + 2 x^{2})$ .

Apply the distributive property.

$10 y^{2} - 6 x^{2} = 66$

$3 (3 y^{2}) + 3 (2 x^{2}) = (3) (35)$

Multiply.

Multiply $3$ by $3$ .

$10 y^{2} - 6 x^{2} = 66$

$9 y^{2} + 3 (2 x^{2}) = (3) (35)$

Multiply $2$ by $3$ .

$10 y^{2} - 6 x^{2} = 66$

$9 y^{2} + 6 x^{2} = (3) (35)$

$10 y^{2} - 6 x^{2} = 66$

$9 y^{2} + 6 x^{2} = (3) (35)$

$10 y^{2} - 6 x^{2} = 66$

$9 y^{2} + 6 x^{2} = (3) (35)$

$10 y^{2} - 6 x^{2} = 66$

$9 y^{2} + 6 x^{2} = (3) (35)$

Simplify the right side.

Multiply $3$ by $35$ .

$10 y^{2} - 6 x^{2} = 66$

$9 y^{2} + 6 x^{2} = 105$

$10 y^{2} - 6 x^{2} = 66$

$9 y^{2} + 6 x^{2} = 105$

$10 y^{2} - 6 x^{2} = 66$

$9 y^{2} + 6 x^{2} = 105$

Step 5

Add the two equations together to eliminate $x^{2}$ from the system.

	$1$	$0$	$y^{2}$	$-$	$6$	$x^{2}$	$=$		$6$	$6$
$+$		$9$	$y^{2}$	$+$	$6$	$x^{2}$	$=$	$1$	$0$	$5$
	$1$	$9$	$y^{2}$				$=$	$1$	$7$	$1$

Step 6

Divide each term in $19 y^{2} = 171$ by $19$ and simplify.

Divide each term in $19 y^{2} = 171$ by $19$ .

$\frac{19 y^{2}}{19} = \frac{171}{19}$

Simplify the left side.

Cancel the common factor of $19$ .

Cancel the common factor.

$\frac{19 y^{2}}{19} = \frac{171}{19}$

Divide $y^{2}$ by $1$ .

$y^{2} = \frac{171}{19}$

Simplify the right side.

Divide $171$ by $19$ .

$y^{2} = 9$

Step 7

Substitute the value found for $y^{2}$ into one of the original equations, then solve for $x^{2}$ .

Substitute the value found for $y^{2}$ into one of the original equations to solve for $x^{2}$ .

$10 (9) - 6 x^{2} = 66$

Multiply $10$ by $9$ .

$90 - 6 x^{2} = 66$

Move all terms not containing $x^{2}$ to the right side of the equation.

Subtract $90$ from both sides of the equation.

$- 6 x^{2} = 66 - 90$

Subtract $90$ from $66$ .

$- 6 x^{2} = - 24$

Divide each term in $- 6 x^{2} = - 24$ by $- 6$ and simplify.

Divide each term in $- 6 x^{2} = - 24$ by $- 6$ .

$\frac{- 6 x^{2}}{- 6} = \frac{- 24}{- 6}$

Simplify the left side.

Cancel the common factor of $- 6$ .

Cancel the common factor.

$\frac{- 6 x^{2}}{- 6} = \frac{- 24}{- 6}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 24}{- 6}$

Simplify the right side.

Divide $- 24$ by $- 6$ .

$x^{2} = 4$

Step 8

This is the final solution to the independent system of equations.

$y^{2} = 9$

$x^{2} = 4$

Step 9

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$y = \pm \sqrt{9}$

$x^{2} = 4$

Step 10

Simplify $\pm \sqrt{9}$ .

Rewrite $9$ as $3^{2}$ .

$y = \pm \sqrt{3^{2}}$

$x^{2} = 4$

Pull terms out from under the radical, assuming positive real numbers.

$y = \pm 3$

$x^{2} = 4$

$y = \pm 3$

$x^{2} = 4$

Step 11

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$y = 3$

$x^{2} = 4$

Next, use the negative value of the $\pm$ to find the second solution.

$y = - 3$

$x^{2} = 4$

The complete solution is the result of both the positive and negative portions of the solution.

$y = 3, - 3$

$x^{2} = 4$

$y = 3, - 3$

$x^{2} = 4$

Step 12

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

$y = 3, - 3$

Step 13

Simplify $\pm \sqrt{4}$ .

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

$y = 3, - 3$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

$y = 3, - 3$

$x = \pm 2$

$y = 3, - 3$

Step 14

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

$y = 3, - 3$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

$y = 3, - 3$

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

$y = 3, - 3$

$x = 2, - 2$

$y = 3, - 3$

Step 15

The final result is the combination of all values of $x$ with all values of $y$ .

$(2, 3), (2, - 3), (- 2, 3), (- 2, - 3)$

Step 16

The result can be shown in multiple forms.

Point Form:

$(2, 3), (2, - 3), (- 2, 3), (- 2, - 3)$

Equation Form:

$x = 2, y = 3$

$x = 2, y = - 3$

$x = - 2, y = 3$

$x = - 2, y = - 3$

Step 17

Solve by Addition/Elimination 3x^2-5y^2=-33 , 2x^2+3y^2=35

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