MHB Solving for U & P in a Coordinate Change

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To solve for U and P in the coordinate transformation z = U⋅x + P, two equations are established based on the given points: -13U + P = 12 and -7U + P = 6. By subtracting the second equation from the first, U can be determined as 3. Substituting U back into either equation allows for the calculation of P, resulting in P being 21. Additionally, to find the original coordinate corresponding to the new coordinate of 11, the equation can be rearranged to solve for x, yielding the original coordinate as -2. The discussion effectively demonstrates how to derive U, P, and the original coordinate using algebraic methods.
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Assume that you are given a coordinate change on a line which changes the coordinate x to a new coordinate z given by the formula z=U⋅x+P where U,P are real numbers with U non zero. If the new coordinate of the point -13 is 12 and the new coordinate of the point -7 is 6 then we must have U= ? and P= ? . Moreover, for the same transformation, if the new coordinate is (11) then the original coordinate must have been ?
 
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You can use the given information to set up a system of equations:

$$-13U+P=12$$

$$-7U+P=6$$
 

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