Final answer:
The system of equations y = x² - 5x + 3 and y = x - 6 has only one solution. The equations are set equal to each other to solve for x, resulting in a single unique solution where x = 3 and the corresponding y value is -3. Therefore, the correct answer is option a) 1.
Step-by-step explanation:
The question asks how many solutions there are for the system of equations given by y = x² - 5x + 3 and y = x - 6. To find the solutions, we can set both equations equal to each other, since they both define y in terms of x, and solve for the possible values of x.
Setting the right-hand sides equal to each other gives us the equation x² - 5x + 3 = x - 6. Rearranging the terms to bring them all to one side gives us x² - 6x + 9 = 0, which is a quadratic equation that simplifies to (x - 3)² = 0. This indicates that there is only one solution for x, which is x = 3. Substituting x = 3 into either of the original equations to find the corresponding y value, we get y = -3. Therefore, there is only one solution to the system of equations, which is (3, -3).
From an algebraic perspective, since the equations represent a parabola and a line, the intersection points of these graphs correspond to the solutions of the system. In this case, the parabola and the line intersect at only one point, thus confirming the one unique solution for this system of equations.