Considering all values of x and y for which x+y is at most 7, x is at least 2, and y is at least -1, what is the minimum value of y-2x?
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graph x+y ≤ 7 AND x ≥ 2 AND y ≥ -1
this results in a set of points forming a triangle with vertices (2,5), (2,0) and (7,0)
let P = -2x + y
sub in those three points shows that (7,0) yields the smallest value of P
so the minimum value of y-2x is -14
the answer is -17 but i don't understand how you get it
I made an arithmetic error,
the 3 points of the triangle should be (2,5),(2,-1) and (8,-1), notice that each point satisfies all 3 of our inequations
so using (8,-1),
y-2x
= -1 - 2(8)
= -17
To determine the minimum value of y-2x, we need to find the values of x and y that satisfy the given conditions and minimize the expression.
Let's start by considering the given conditions:
1. x+y is at most 7: This constraint implies that x+y can't exceed 7.
2. x is at least 2: This constraint means x must be greater than or equal to 2.
3. y is at least -1: This constraint means y must be greater than or equal to -1.
We can use these constraints to find the possible range of values for x and y.
For the first condition, x+y ≤ 7, we can rewrite it as y ≤ 7-x. This equation represents a line with a slope of -1 and y-intercept of 7.
For the second and third conditions, x ≥ 2 and y ≥ -1, we can plot these as vertical and horizontal lines, respectively.
By examining the graph formed by these lines, we can identify the feasible region that satisfies all the given conditions.
Now, let's find the minimum value of y-2x within this feasible region.
One possible approach is to evaluate the expression y-2x at the vertices (corners) of the feasible region and choose the minimum value obtained.
So, we need to find the coordinates of the vertices where the lines intersect.
From the graph, we observe that the vertices of interest are (2, 5), (2, -1), and (6, -1).
Calculating y-2x at each vertex:
- For (2, 5): y-2x = 5 - 2(2) = 5 - 4 = 1
- For (2, -1): y-2x = (-1) - 2(2) = -1 - 4 = -5
- For (6, -1): y-2x = (-1) - 2(6) = -1 - 12 = -13
Therefore, the minimum value of y-2x is -13.
In conclusion, the minimum value of y-2x, considering all values of x and y for which x+y is at most 7, x is at least 2, and y is at least -1, is -13.