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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ok i think i see now thank you!
To find the minimum value of y - 2x, we need to first determine the range of x and y that satisfy the given conditions:
1. x is at least 2: This means x ≥ 2.
2. y is at least -1: This means y ≥ -1.
3. x + y is at most 7: This means x + y ≤ 7.
Combining the first and third conditions, we get:
x + y ≤ 7
x + (y - 2x) ≤ 7
-y + x ≤ 7
x - y ≥ -7 (by multiplying both sides by -1 and reversing the inequality sign)
Now, we need to find the range of values of x and y that satisfy the above condition.
Since x is at least 2, we can select any value of x greater than or equal to 2.
For simplicity, let's choose x = 2. Substituting this into the inequality, we get:
2 - y ≥ -7
-y ≥ -7 - 2
-y ≥ -9 (by simplifying)
Now, we multiply both sides of the inequality by -1 (and reverse the inequality sign) to solve for y:
y ≤ 9
So, for x = 2, the range of values for y that satisfies the conditions is y ≤ 9.
Therefore, the minimum value of y - 2x occurs when x = 2 and y = 9, and the minimum value is:
y - 2x = 9 - 2(2) = 9 - 4 = 5.
Hence, the minimum value of y - 2x is 5.