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Solve for y after substituting into a two-variable function

Treat the remaining variable as a quadratic unknown

In $f (x, y) = c$ , substitute the given $x$ first. Collect terms in $y$ . If the result is quadratic, solve for both roots and check both in the original function.

Problem

If $f (x, y) = 3 x^{2} - 4 x y + 5 y^{2}$ and $f (4, y) = 100$ , find every possible value of $y$ .

Answer

y = \frac{26}{5} = 5.2

y = - 2

Both values make f(4, y) equal 100.

Step-by-step solution

1. Substitute x = 4

3 (4)^{2} - 4 (4) y + 5 y^{2} = 100,

which becomes $5 y^{2} - 16 y - 52 = 0$ .

2. Solve the quadratic

The discriminant is $(- 16)^{2} - 4 (5) (- 52) = 1296 = 3 6^{2}$ . Therefore

y = \frac{16 \pm 36}{10},

giving $y = 52/10 = 26/5$ or $y = - 20/10 = - 2$ .

3. Check

For $y = 26/5$ , $48 - 16 y + 5 y^{2} = 100$ . For $y = - 2$ , $48 + 32 + 20 = 100$ . Both roots are valid.

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