Determine the number of real solutions:
$4y - 7 = 5x^2 - x + 2 + 3y$
□ real solution(s)
The given equation is \(4y - 7 = 5x^2 - x + 2 + 3y\). To determine the number of real solutions \((x, y)\), we first isolate \(y\) to express it in terms of \(x\). By subtracting \(3y\) from both sides, we get:
\[(4y - 3y) - 7 = 5x^2 - x + 2\]
\[y - 7 = 5x^2 - x + 2\]
Adding 7 to both sides, we obtain the function:
\[y = 5x^2 - x + 9\]
The equation \(y = 5x^2 - x + 9\) defines \(y\) as a function of \(x\). This is a quadratic function, representing a parabola in the coordinate plane. For every real number chosen for \(x\), we can substitute it into the expression \(5x^2 - x + 9\) to calculate a unique corresponding real value for \(y\). Since the domain of \(x\) is the set of all real numbers, and for each \(x\) there exists exactly one \(y\), there is a distinct pair \((x, y)\) for every real number \(x\). Because there are infinitely many real numbers, there are infinitely many such pairs \((x, y)\) that satisfy the original equation. Therefore, the number of real solutions is infinite.
This type of problem relates to the concept of dependent variables and functional relationships. In real-life scenarios, such as predicting the cost of producing an item based on the number of units manufactured, one variable (the cost) is often a function of another (the quantity). By identifying that an equation represents a function, we understand that for every possible input (quantity), there is a determined output (cost). In our case, the relationship \(y = 5x^2 - x + 9\) shows how one variable scales with the other, confirming that there isn't just one solution, but a continuous range of possibilities defined by the input variable.