• In this problem, we are moving from point P, to point P', and then to point P''.
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• To move from one point to another, we make translations of the points.
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• The ticks of the points are simply a notation. One tick denotes the point where you are after the first translation. Two ticks denote the point where you are after the second translation.
• Mathematically, a translation from a point P to a new point P' consists in summing numbers to the coordinates of the point P to get the coordinates of the new point P'.
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• For example, if we have the point P with coordinates (x,y), and we move that point 3 units to the right, and 4 units up, we get the point P' with coordinates:
[tex]P^{\prime}=(x+3,y+4)\text{.}[/tex]
This problem consists of the following:
1) We start with a point P(4,-2) with coordinates x = 4 and y = -2.
2) We make the first translation, which consists in going from point P to point P'.
The coordinates of point P' are given by:
[tex]P^{\prime}=(x+3,y-a).[/tex]
The coordinates x and y in that formula are the values of x and y of the original point. So we must replace x = 4 and y = -2 in the formula above. Doing that we have the following coordinates for the point P':
[tex]P^{\prime}=(4+3,-2-a)=(7,-2-a)\text{.}[/tex]
3) We made the first translation. Now we will do another translation, from point P' with coordinates x = 7 and y = -2 - a, to the point P'' with coordinates:
[tex]P^{\prime\prime}=(x-b,y+7).[/tex]
Replacing the values x = 7 and y = -2 - a we get the following coordinates for P'':
[tex]\begin{gathered} P^{\prime\prime}=(7-b,-2-a+7), \\ P^{\prime\prime}=(7-b,5-a), \end{gathered}[/tex]
4) Finally, doing the translations we get the following coordinates for point P'':
[tex]P^{\prime\prime}=(7-b,5-a)\text{.}[/tex]
But from the statement of the problem, we know that the coordinates of point P'' are:
[tex]P^{\prime\prime}=(-5,8)\text{.}[/tex]
Comparing each coordinate we have the following equations:
[tex]7-b=-5,\text{ and }^{}5-a=8.[/tex]
Solving the equation of b, we get:
[tex]\begin{gathered} 7-b=-5, \\ 7=-5+b, \\ b=7+5, \\ b=12. \end{gathered}[/tex]
Solving the equation of a, we get:
[tex]\begin{gathered} 5-a=8, \\ 5=8+a, \\ a=5-8, \\ a=-3. \end{gathered}[/tex]
So the values of a and b are:
[tex]\begin{gathered} a=-3, \\ b=12. \end{gathered}[/tex]
5) Using the value a = -3, the coordinates of point P' are:
[tex]P^{\prime}=(7,-2-(-3))=(7,-2+3)=(7,1)\text{.}[/tex]
Answers
[tex]\begin{gathered} a=-3, \\ b=12, \\ P^{\prime}(7,1)\text{.} \end{gathered}[/tex]
