Final answer:
The value of w can be any real number since, for x=1, the first equation simplifies to a line with a non-zero slope, which will intersect the horizontal line represented by the second equation exactly once.
Step-by-step explanation:
The question asks for the value of w for which the system of equations will have exactly one solution for x=1. Given the first equation is wx + 2y = 3(1 + y) and we know that x=1, we can substitute 1 for x to simplify the equation:
w(1) + 2y = 3 + 3y ⇒ w + 2y = 3 + 3y
To have a unique solution, we need these two equations to not be parallel, meaning their slopes must be different. In this case, we isolate y in both equations and compare their slopes. The first equation can be rearranged to get y on one side:
w + 2y = 3 + 3y ⇒ 2y - 3y = 3 - w ⇒ -y = 3 - w ⇒ y = w - 3
The slope of the equation y = w - 3 is 1, since it is in the form y = mx + b, where m is the slope. However, to ensure this equation has a unique solution with the second equation 18 - y = 2(1 - y) + 3, we must also consider the slope of this equation. Simplifying the second equation yields:
18 - y = 2 - 2y + 3 ⇒ 18 - y = 5 - 2y ⇒ y = 13
Here, the slope is 0 because the equation represents a horizontal line. Therefore, any non-zero slope in the first equation will ensure that it is not parallel to the second equation, giving us a unique solution. Thus, w can take any value except for the value that would make the slope of the first equation equal to 0.
However, since the second equation is a constant function (slope is 0), any value of w that ensures the first equation has a non-zero slope will result in a single intersection point. Since a slope of 1 is already non-zero, any value of w will give a unique solution with the second equation.