Question

7-14 Determine whether the equation or table defines y as a function of x.

\({\bf{2}}xy + {\bf{5}}{y^{\bf{2}}} = {\bf{4}}\)

Short answer

The equation does not define y as a function of x.

Step-by-step solution

Simplify the given equation

The equation \(2xy + 5{y^2} = 4\) can be simplified as

\(5{y^2} + \left( {2x} \right)y - 4 = 0\).

Find the root of the quadratic equation

The root of the equation \(5{y^2} + \left( {2x} \right)y - 4 = 0\)is determined below:

\(\begin{aligned}y &= \frac{{ - 2x \pm \sqrt {4{x^2} - 4\left( 5 \right)\left( { - 4} \right)} }}{{2\left( 5 \right)}}\\ &= \frac{{ - 2x \pm \sqrt {4{x^2} + 80} }}{{10}}\\ &= \frac{{ - x \pm \sqrt {{x^2} + 20} }}{5}\end{aligned}\)

From the above equation, for one value of x, there is more than one value of y.

So, the given equation does not define y as a function of x.