For each equation, (a) solve for x in terms of y. See Example 8. ...

Question

For each equation, (a) solve for x in terms of y. See Example 8. 2x^2 + 4xy - 3y^2 = 2

Accepted Answer

Hey, everyone here, we are asked to express X in terms of Y in the following equation, nine X squared minus 15 X Y plus five Y squared is equal to seven. Now, our first step here is to rearrange our equation so that it is equal to zero. So that means we just need to move this positive seven for from the right side to the left side. So doing this, we have nine X squared minus 15 X Y plus five Y squared minus seven is equal to zero. And now looking at the equation we have so far, we see that it follows the quadratic equation formula which is A X squared plus B X plus C is equal to zero. So now we see we can identify our values for A B and C by comparing this formula with our equation. So we will treat X as our variable here since we are expressing our equation in terms or S X in terms of why. So why will be treated as a constant? And therefore, we can start by finding our value for A and A is the coefficient of X squared. So A is just nine, our value for B is going to be the coefficient for X. So we see that the coefficient is negative 15 Y and then our value for C are the remaining constants. And again, we're treating Y as a constant. So C is going to be five Y squared minus seven. And so now that we have identified our values for A B and C, we can apply the quadratic formula. Well, first we need to recall the quadratic formula which is in terms of X, we have X is equal to neck B plus or minus the square root of B squared minus four A C all divided by two A. And again, we have our A B and C values identified. So we can go ahead and just substitute them into our quadratic formula. So doing this, we have X is equal to the negative quantity of negative 15 Y plus or minus the square root of the quantity of negative 15 Y squared minus four times nine times the quantity of five Y squared minus seven. And this is all divided by two times nine. And now we just need to continue to simplify this expression. So doing this, we have first positive 15 Y plus or minus the square root of Y squared minus 36 times the quantity of five Y squared minus seven. And this again is all divided by two times nine, which simplifies to just 18 Now continuing to simplify our route here by factoring in this neck 36 we will have 15 Y plus or minus the square root of 225 Y squared minus 180 Y squared plus 252 all divided by 18. And now we just have to finish simplifying by combine our like terms underneath the root. And we will see here that X is equal to Y plus or minus the square root of 225 minus 180 gives us 45 Y squared plus 252 all divided by 18. So now we have simplified our expression and more importantly, expressed our original equation as X in terms of Y. Therefore, we are done solving this problem and we are left with answer choice. A thanks for watching. I hope you found this video helpful and I'll see you again next time.

For each equation, (a) solve for x in terms of y. See Example 8. 2x^2 + 4xy - 3y^2 = 2

Key Concepts

Quadratic Equations

Solving for a Variable

The Quadratic Formula

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