Solve each nonlinear system of equations. Give all solutions, ...

Question

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.

$x^{2} + y^{2} = 5$

$- 3 x + 4 y = 2$

Accepted Answer

Hey, everyone. Here we are asked to provide all solutions for the given nonlinear system of equations including non-real complex components. Here we are given two equations. The first one being X squared plus Y squared is equal equal to one. And the second equation is five X plus Y is equal to five. Here we have four answer choice options, answer A through D. Each of them being a solution set containing two different ordered pairs. So here we can utilize the method of substitution to solve for X initially. So we can look at our second equation and easily solve four Y by first just subtracting five X from both sides. So we have Y is equal to five minus five X. Now we can just take this expression we found four Y and substitute it into our first equation. Recalling our first equation, we have X squared plus Y squared is equal to one. And now substituting in our new expression, we have X squared plus the quantity of five minus five X squared and this is still equal to one. Now, we just want to expand our squared expression. So we have X squared plus 25 minus 50 X plus 25 X squared is equal to one. Now, just combining like terms, we have 26 X squared minus 50 X plus 24 is equal to zero. And so now we just want to factor this expression on the left side. So we have the quantity of X minus one times the quantity of 13 X minus 12 and this is equal to zero. Now, recalling the zero factor property, we can take each individual expression in each set of parentheses and set them equal to zero. So starting with the first expression, we have X minus one. And again, we set this equal to zero. And now we can solve for X by simply adding one to both sides. So we have X is equal to one. Now we can repeat this with the other part of our expression where we have 13 X minus 12. Again, setting this expression equal to zero. Now solving for X, we first add 12 to both sides. So we have 13, X is equal to 12. Now we just need to divide both sides by 13. And we see that our other value for X is 12 divided by 13. So now we want to take each of these values for X and substitute them into our expression we found for Y. So instead of having Y is equal to five minus five, X, we can begin by substituting in our first value for X. Which was one. So we have Y is equal to five minus five times one. Now, simplifying we have Y is equal to five minus five. And so we see that one of our solutions is for when X is equal to one Y is equal to zero. So now we can use the other value of X and repeat the same process. So we have Y is equal to five minus five times 12 divided by 13. So now just simplifying, we have Y is equal to five minus 60 divided by 13. And now simplifying further, we have our other value for Y as five divided by 13. So when X is equal to 12, divided by 13, Y is then equal to five divided by 13. So now with all of our X and Y values, we can rewrite our solution as these solutions set where we have one ordered pair as one and zero and then our other ordered pair is 12/13 and 5/13. So with these two ordered pairs rewritten in a solution set, we are left with answer choice. A where again, we have the solution set of two ordered pairs, one of them being one and zero and the other is 12/13 and 5/13. Thanks for watching. I hope you found this video helpful and I'll see you again next time.

Solve each nonlinear system of equations. Give all solutions, including those with nonreal complex components. See Examples 1–5.
$x^{2} + y^{2} = 5$
$- 3 x + 4 y = 2$

Key Concepts

Nonlinear Systems of Equations

Substitution Method

Complex Solutions

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