Final answer:
In equation a, y is not uniquely determined by x, while in equations b and c, y is uniquely determined by x. In equation d, y is not uniquely determined by x.
Step-by-step explanation:
a. x²−7y=3: This equation does not define y as a function of x because for a given x, there are two possible values for y. For example, if we let x = 4, then the equation becomes 16−7y=3, which can be rearranged to find that y could be either 2 or -2. Therefore, y is not uniquely determined by x, and the equation does not define y as a function of x.
b. 2x+4y=8: This equation does define y as a function of x. We can solve for y in terms of x by solving for y using algebraic manipulation. First, subtract 2x from both sides to get 4y = -2x + 8, then divide by 4 to get y = -0.5x + 2. Therefore, y is uniquely determined by x, and the equation defines y as a function of x.
c. y²=x+1: This equation does define y as a function of x. By solving for y, we can see that y = ±√(x+1). Although there are two possible roots, the value of y is still uniquely determined by the value of x. Therefore, the equation defines y as a function of x.
d. 3x−2y²=5: This equation does not define y as a function of x because for a given x, there are two possible values for y. For example, if we let x = 2, then the equation becomes 6−2y²=5, which can be rearranged to find that y could be either 1 or -1. Therefore, y is not uniquely determined by x, and the equation does not define y as a function of x.