Find the Points of Intersection 8x^2+5y^2=40 , x=y+5

$8 x^{2} + 5 y^{2} = 40$ , $x = y + 5$

Step 1

Replace all occurrences of $x$ with $y + 5$ in each equation.

Replace all occurrences of $x$ in $8 x^{2} + 5 y^{2} = 40$ with $y + 5$ .

$8 {(y + 5)}^{2} + 5 y^{2} = 40$

$x = y + 5$

Simplify the left side.

Simplify $8 {(y + 5)}^{2} + 5 y^{2}$ .

Simplify each term.

Rewrite ${(y + 5)}^{2}$ as $(y + 5) (y + 5)$ .

$8 ((y + 5) (y + 5)) + 5 y^{2} = 40$

$x = y + 5$

Expand $(y + 5) (y + 5)$ using the FOIL Method.

Apply the distributive property.

$8 (y (y + 5) + 5 (y + 5)) + 5 y^{2} = 40$

$x = y + 5$

Apply the distributive property.

$8 (y \cdot y + y \cdot 5 + 5 (y + 5)) + 5 y^{2} = 40$

$x = y + 5$

Apply the distributive property.

$8 (y \cdot y + y \cdot 5 + 5 y + 5 \cdot 5) + 5 y^{2} = 40$

$x = y + 5$

$8 (y \cdot y + y \cdot 5 + 5 y + 5 \cdot 5) + 5 y^{2} = 40$

$x = y + 5$

Simplify and combine like terms.

Simplify each term.

Multiply $y$ by $y$ .

$8 (y^{2} + y \cdot 5 + 5 y + 5 \cdot 5) + 5 y^{2} = 40$

$x = y + 5$

Move $5$ to the left of $y$ .

$8 (y^{2} + 5 \cdot y + 5 y + 5 \cdot 5) + 5 y^{2} = 40$

$x = y + 5$

Multiply $5$ by $5$ .

$8 (y^{2} + 5 y + 5 y + 25) + 5 y^{2} = 40$

$x = y + 5$

$8 (y^{2} + 5 y + 5 y + 25) + 5 y^{2} = 40$

$x = y + 5$

Add $5 y$ and $5 y$ .

$8 (y^{2} + 10 y + 25) + 5 y^{2} = 40$

$x = y + 5$

$8 (y^{2} + 10 y + 25) + 5 y^{2} = 40$

$x = y + 5$

Apply the distributive property.

$8 y^{2} + 8 (10 y) + 8 \cdot 25 + 5 y^{2} = 40$

$x = y + 5$

Simplify.

Multiply $10$ by $8$ .

$8 y^{2} + 80 y + 8 \cdot 25 + 5 y^{2} = 40$

$x = y + 5$

Multiply $8$ by $25$ .

$8 y^{2} + 80 y + 200 + 5 y^{2} = 40$

$x = y + 5$

$8 y^{2} + 80 y + 200 + 5 y^{2} = 40$

$x = y + 5$

$8 y^{2} + 80 y + 200 + 5 y^{2} = 40$

$x = y + 5$

Add $8 y^{2}$ and $5 y^{2}$ .

$13 y^{2} + 80 y + 200 = 40$

$x = y + 5$

$13 y^{2} + 80 y + 200 = 40$

$x = y + 5$

$13 y^{2} + 80 y + 200 = 40$

$x = y + 5$

$13 y^{2} + 80 y + 200 = 40$

$x = y + 5$

Step 2

Solve for $y$ in $13 y^{2} + 80 y + 200 = 40$ .

Move all terms to the left side of the equation and simplify.

Subtract $40$ from both sides of the equation.

$13 y^{2} + 80 y + 200 - 40 = 0$

$x = y + 5$

Subtract $40$ from $200$ .

$13 y^{2} + 80 y + 160 = 0$

$x = y + 5$

$13 y^{2} + 80 y + 160 = 0$

$x = y + 5$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

$x = y + 5$

Substitute the values $a = 13$ , $b = 80$ , and $c = 160$ into the quadratic formula and solve for $y$ .

$\frac{- 80 \pm \sqrt{80^{2} - 4 \cdot (13 \cdot 160)}}{2 \cdot 13}$

$x = y + 5$

Simplify.

Simplify the numerator.

Raise $80$ to the power of $2$ .

$y = \frac{- 80 \pm \sqrt{6400 - 4 \cdot 13 \cdot 160}}{2 \cdot 13}$

$x = y + 5$

Multiply $- 4 \cdot 13 \cdot 160$ .

Multiply $- 4$ by $13$ .

$y = \frac{- 80 \pm \sqrt{6400 - 52 \cdot 160}}{2 \cdot 13}$

$x = y + 5$

Multiply $- 52$ by $160$ .

$y = \frac{- 80 \pm \sqrt{6400 - 8320}}{2 \cdot 13}$

$x = y + 5$

$y = \frac{- 80 \pm \sqrt{6400 - 8320}}{2 \cdot 13}$

$x = y + 5$

Subtract $8320$ from $6400$ .

$y = \frac{- 80 \pm \sqrt{- 1920}}{2 \cdot 13}$

$x = y + 5$

Rewrite $- 1920$ as $- 1 (1920)$ .

$y = \frac{- 80 \pm \sqrt{- 1 \cdot 1920}}{2 \cdot 13}$

$x = y + 5$

Rewrite $\sqrt{- 1 (1920)}$ as $\sqrt{- 1} \cdot \sqrt{1920}$ .

$y = \frac{- 80 \pm \sqrt{- 1} \cdot \sqrt{1920}}{2 \cdot 13}$

$x = y + 5$

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 80 \pm i \cdot \sqrt{1920}}{2 \cdot 13}$

$x = y + 5$

Rewrite $1920$ as $8^{2} \cdot 30$ .

Factor $64$ out of $1920$ .

$y = \frac{- 80 \pm i \cdot \sqrt{64 (30)}}{2 \cdot 13}$

$x = y + 5$

Rewrite $64$ as $8^{2}$ .

$y = \frac{- 80 \pm i \cdot \sqrt{8^{2} \cdot 30}}{2 \cdot 13}$

$x = y + 5$

$y = \frac{- 80 \pm i \cdot \sqrt{8^{2} \cdot 30}}{2 \cdot 13}$

$x = y + 5$

Pull terms out from under the radical.

$y = \frac{- 80 \pm i \cdot (8 \sqrt{30})}{2 \cdot 13}$

$x = y + 5$

Move $8$ to the left of $i$ .

$y = \frac{- 80 \pm 8 i \sqrt{30}}{2 \cdot 13}$

$x = y + 5$

$y = \frac{- 80 \pm 8 i \sqrt{30}}{2 \cdot 13}$

$x = y + 5$

Multiply $2$ by $13$ .

$y = \frac{- 80 \pm 8 i \sqrt{30}}{26}$

$x = y + 5$

Simplify $\frac{- 80 \pm 8 i \sqrt{30}}{26}$ .

$y = \frac{- 40 \pm 4 i \sqrt{30}}{13}$

$x = y + 5$

$y = \frac{- 40 \pm 4 i \sqrt{30}}{13}$

$x = y + 5$

Simplify the expression to solve for the $+$ portion of the $\pm$ .

Simplify the numerator.

Raise $80$ to the power of $2$ .

$y = \frac{- 80 \pm \sqrt{6400 - 4 \cdot 13 \cdot 160}}{2 \cdot 13}$

$x = y + 5$

Multiply $- 4 \cdot 13 \cdot 160$ .

Multiply $- 4$ by $13$ .

$y = \frac{- 80 \pm \sqrt{6400 - 52 \cdot 160}}{2 \cdot 13}$

$x = y + 5$

Multiply $- 52$ by $160$ .

$y = \frac{- 80 \pm \sqrt{6400 - 8320}}{2 \cdot 13}$

$x = y + 5$

$y = \frac{- 80 \pm \sqrt{6400 - 8320}}{2 \cdot 13}$

$x = y + 5$

Subtract $8320$ from $6400$ .

$y = \frac{- 80 \pm \sqrt{- 1920}}{2 \cdot 13}$

$x = y + 5$

Rewrite $- 1920$ as $- 1 (1920)$ .

$y = \frac{- 80 \pm \sqrt{- 1 \cdot 1920}}{2 \cdot 13}$

$x = y + 5$

Rewrite $\sqrt{- 1 (1920)}$ as $\sqrt{- 1} \cdot \sqrt{1920}$ .

$y = \frac{- 80 \pm \sqrt{- 1} \cdot \sqrt{1920}}{2 \cdot 13}$

$x = y + 5$

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 80 \pm i \cdot \sqrt{1920}}{2 \cdot 13}$

$x = y + 5$

Rewrite $1920$ as $8^{2} \cdot 30$ .

Factor $64$ out of $1920$ .

$y = \frac{- 80 \pm i \cdot \sqrt{64 (30)}}{2 \cdot 13}$

$x = y + 5$

Rewrite $64$ as $8^{2}$ .

$y = \frac{- 80 \pm i \cdot \sqrt{8^{2} \cdot 30}}{2 \cdot 13}$

$x = y + 5$

$y = \frac{- 80 \pm i \cdot \sqrt{8^{2} \cdot 30}}{2 \cdot 13}$

$x = y + 5$

Pull terms out from under the radical.

$y = \frac{- 80 \pm i \cdot (8 \sqrt{30})}{2 \cdot 13}$

$x = y + 5$

Move $8$ to the left of $i$ .

$y = \frac{- 80 \pm 8 i \sqrt{30}}{2 \cdot 13}$

$x = y + 5$

$y = \frac{- 80 \pm 8 i \sqrt{30}}{2 \cdot 13}$

$x = y + 5$

Multiply $2$ by $13$ .

$y = \frac{- 80 \pm 8 i \sqrt{30}}{26}$

$x = y + 5$

Simplify $\frac{- 80 \pm 8 i \sqrt{30}}{26}$ .

$y = \frac{- 40 \pm 4 i \sqrt{30}}{13}$

$x = y + 5$

Change the $\pm$ to $+$ .

$y = \frac{- 40 + 4 i \sqrt{30}}{13}$

$x = y + 5$

Rewrite $- 40$ as $- 1 (40)$ .

$y = \frac{- 1 \cdot 40 + 4 i \sqrt{30}}{13}$

$x = y + 5$

Factor $- 1$ out of $4 i \sqrt{30}$ .

$y = \frac{- 1 \cdot 40 - (- 4 i \sqrt{30})}{13}$

$x = y + 5$

Factor $- 1$ out of $- 1 (40) - (- 4 i \sqrt{30})$ .

$y = \frac{- 1 (40 - 4 i \sqrt{30})}{13}$

$x = y + 5$

Move the negative in front of the fraction.

$y = - \frac{40 - 4 i \sqrt{30}}{13}$

$x = y + 5$

$y = - \frac{40 - 4 i \sqrt{30}}{13}$

$x = y + 5$

Simplify the expression to solve for the $-$ portion of the $\pm$ .

Simplify the numerator.

Raise $80$ to the power of $2$ .

$y = \frac{- 80 \pm \sqrt{6400 - 4 \cdot 13 \cdot 160}}{2 \cdot 13}$

$x = y + 5$

Multiply $- 4 \cdot 13 \cdot 160$ .

Multiply $- 4$ by $13$ .

$y = \frac{- 80 \pm \sqrt{6400 - 52 \cdot 160}}{2 \cdot 13}$

$x = y + 5$

Multiply $- 52$ by $160$ .

$y = \frac{- 80 \pm \sqrt{6400 - 8320}}{2 \cdot 13}$

$x = y + 5$

$y = \frac{- 80 \pm \sqrt{6400 - 8320}}{2 \cdot 13}$

$x = y + 5$

Subtract $8320$ from $6400$ .

$y = \frac{- 80 \pm \sqrt{- 1920}}{2 \cdot 13}$

$x = y + 5$

Rewrite $- 1920$ as $- 1 (1920)$ .

$y = \frac{- 80 \pm \sqrt{- 1 \cdot 1920}}{2 \cdot 13}$

$x = y + 5$

Rewrite $\sqrt{- 1 (1920)}$ as $\sqrt{- 1} \cdot \sqrt{1920}$ .

$y = \frac{- 80 \pm \sqrt{- 1} \cdot \sqrt{1920}}{2 \cdot 13}$

$x = y + 5$

Rewrite $\sqrt{- 1}$ as $i$ .

$y = \frac{- 80 \pm i \cdot \sqrt{1920}}{2 \cdot 13}$

$x = y + 5$

Rewrite $1920$ as $8^{2} \cdot 30$ .

Factor $64$ out of $1920$ .

$y = \frac{- 80 \pm i \cdot \sqrt{64 (30)}}{2 \cdot 13}$

$x = y + 5$

Rewrite $64$ as $8^{2}$ .

$y = \frac{- 80 \pm i \cdot \sqrt{8^{2} \cdot 30}}{2 \cdot 13}$

$x = y + 5$

$y = \frac{- 80 \pm i \cdot \sqrt{8^{2} \cdot 30}}{2 \cdot 13}$

$x = y + 5$

Pull terms out from under the radical.

$y = \frac{- 80 \pm i \cdot (8 \sqrt{30})}{2 \cdot 13}$

$x = y + 5$

Move $8$ to the left of $i$ .

$y = \frac{- 80 \pm 8 i \sqrt{30}}{2 \cdot 13}$

$x = y + 5$

$y = \frac{- 80 \pm 8 i \sqrt{30}}{2 \cdot 13}$

$x = y + 5$

Multiply $2$ by $13$ .

$y = \frac{- 80 \pm 8 i \sqrt{30}}{26}$

$x = y + 5$

Simplify $\frac{- 80 \pm 8 i \sqrt{30}}{26}$ .

$y = \frac{- 40 \pm 4 i \sqrt{30}}{13}$

$x = y + 5$

Change the $\pm$ to $-$ .

$y = \frac{- 40 - 4 i \sqrt{30}}{13}$

$x = y + 5$

Rewrite $- 40$ as $- 1 (40)$ .

$y = \frac{- 1 \cdot 40 - 4 i \sqrt{30}}{13}$

$x = y + 5$

Factor $- 1$ out of $- 4 i \sqrt{30}$ .

$y = \frac{- 1 \cdot 40 - (4 i \sqrt{30})}{13}$

$x = y + 5$

Factor $- 1$ out of $- 1 (40) - (4 i \sqrt{30})$ .

$y = \frac{- 1 (40 + 4 i \sqrt{30})}{13}$

$x = y + 5$

Move the negative in front of the fraction.

$y = - \frac{40 + 4 i \sqrt{30}}{13}$

$x = y + 5$

$y = - \frac{40 + 4 i \sqrt{30}}{13}$

$x = y + 5$

The final answer is the combination of both solutions.

$y = - \frac{40 - 4 i \sqrt{30}}{13}, - \frac{40 + 4 i \sqrt{30}}{13}$

$x = y + 5$

$y = - \frac{40 - 4 i \sqrt{30}}{13}, - \frac{40 + 4 i \sqrt{30}}{13}$

$x = y + 5$

Step 3

Replace all occurrences of $y$ with $- \frac{40 - 4 i \sqrt{30}}{13}$ in each equation.

Replace all occurrences of $y$ in $x = y + 5$ with $- \frac{40 - 4 i \sqrt{30}}{13}$ .

$x = (- \frac{40 - 4 i \sqrt{30}}{13}) + 5$

$y = - \frac{40 - 4 i \sqrt{30}}{13}$

Simplify the right side.

Simplify $(- \frac{40 - 4 i \sqrt{30}}{13}) + 5$ .

To write $5$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$x = - \frac{40 - 4 i \sqrt{30}}{13} + 5 \cdot \frac{13}{13}$

$y = - \frac{40 - 4 i \sqrt{30}}{13}$

Combine $5$ and $\frac{13}{13}$ .

$x = - \frac{40 - 4 i \sqrt{30}}{13} + \frac{5 \cdot 13}{13}$

$y = - \frac{40 - 4 i \sqrt{30}}{13}$

Combine the numerators over the common denominator.

$x = \frac{25 + 4 i \sqrt{30}}{13}$

$y = - \frac{40 - 4 i \sqrt{30}}{13}$

$x = \frac{25 + 4 i \sqrt{30}}{13}$

$y = - \frac{40 - 4 i \sqrt{30}}{13}$

$x = \frac{25 + 4 i \sqrt{30}}{13}$

$y = - \frac{40 - 4 i \sqrt{30}}{13}$

$x = \frac{25 + 4 i \sqrt{30}}{13}$

$y = - \frac{40 - 4 i \sqrt{30}}{13}$

Step 4

Replace all occurrences of $y$ with $- \frac{40 + 4 i \sqrt{30}}{13}$ in each equation.

Replace all occurrences of $y$ in $x = y + 5$ with $- \frac{40 + 4 i \sqrt{30}}{13}$ .

$x = (- \frac{40 + 4 i \sqrt{30}}{13}) + 5$

$y = - \frac{40 + 4 i \sqrt{30}}{13}$

Simplify the right side.

Simplify $(- \frac{40 + 4 i \sqrt{30}}{13}) + 5$ .

To write $5$ as a fraction with a common denominator, multiply by $\frac{13}{13}$ .

$x = - \frac{40 + 4 i \sqrt{30}}{13} + 5 \cdot \frac{13}{13}$

$y = - \frac{40 + 4 i \sqrt{30}}{13}$

Combine $5$ and $\frac{13}{13}$ .

$x = - \frac{40 + 4 i \sqrt{30}}{13} + \frac{5 \cdot 13}{13}$

$y = - \frac{40 + 4 i \sqrt{30}}{13}$

Combine the numerators over the common denominator.

$x = \frac{25 - 4 i \sqrt{30}}{13}$

$y = - \frac{40 + 4 i \sqrt{30}}{13}$

$x = \frac{25 - 4 i \sqrt{30}}{13}$

$y = - \frac{40 + 4 i \sqrt{30}}{13}$

$x = \frac{25 - 4 i \sqrt{30}}{13}$

$y = - \frac{40 + 4 i \sqrt{30}}{13}$

$x = \frac{25 - 4 i \sqrt{30}}{13}$

$y = - \frac{40 + 4 i \sqrt{30}}{13}$

Step 5

List all of the solutions.

$x = \frac{25 + 4 i \sqrt{30}}{13}, y = - \frac{40 - 4 i \sqrt{30}}{13}$

$x = \frac{25 - 4 i \sqrt{30}}{13}, y = - \frac{40 + 4 i \sqrt{30}}{13}$

Step 6

Find the Points of Intersection 8x^2+5y^2=40 , x=y+5

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