Rotation - <u>Rotation</u> is a plane transformation that rotates every point in a figure through a given angle and direction on a fixed point.
Transformation - <u>Transformation</u> is a process that alters the shape and or position of a point, a line, or a geometric figure.
Translation - <u>Translation</u> occurs when a figure is relocated from one position to another without changing its size, shape, or orientation.
Dilation - <u>Dilation</u> is a transformation that alters the size of a figure, whether enlarging or shrinking it, but it has no effect on the shape of the figure.
Angle of Rotation - <u>Angle of Rotation</u> is the amount that a figure is rotated about a fixed point referred to as the point of rotation.
Line of Reflection - <u>Line of Reflection</u> is a transformation in which each point in a shape appears at an equal distance on the opposite side of a given line.
<em>Hope this helps!!</em>
Given:
Three equations are
![6(0.5x-1.5)+2x=-9-(x+6)](https://tex.z-dn.net/?f=6%280.5x-1.5%29%2B2x%3D-9-%28x%2B6%29)
![4(0.5x-2)+3x=-14-(x+6)](https://tex.z-dn.net/?f=4%280.5x-2%29%2B3x%3D-14-%28x%2B6%29)
![8(0.5x-3)+4x=-9-(x+6)](https://tex.z-dn.net/?f=8%280.5x-3%29%2B4x%3D-9-%28x%2B6%29)
Three solutions are
.
To find:
The solution of each equation from the given solutions.
Solution:
We have,
![6(0.5x-1.5)+2x=-9-(x+6)](https://tex.z-dn.net/?f=6%280.5x-1.5%29%2B2x%3D-9-%28x%2B6%29)
Using distributive property, we get
![3x-9+2x=-9-x-6](https://tex.z-dn.net/?f=3x-9%2B2x%3D-9-x-6)
![5x-9=-15-x](https://tex.z-dn.net/?f=5x-9%3D-15-x)
Isolate variable terms.
![5x+x=-15+9](https://tex.z-dn.net/?f=5x%2Bx%3D-15%2B9)
![6x=-6](https://tex.z-dn.net/?f=6x%3D-6)
Divide both sides by 6.
![x=-1](https://tex.z-dn.net/?f=x%3D-1)
Similarly,
![4(0.5x-2)+3x=-14-(x+6)](https://tex.z-dn.net/?f=4%280.5x-2%29%2B3x%3D-14-%28x%2B6%29)
![2x-8+3x=-14-x-6](https://tex.z-dn.net/?f=2x-8%2B3x%3D-14-x-6)
![5x-8=-20-x](https://tex.z-dn.net/?f=5x-8%3D-20-x)
![5x+x=-20+8](https://tex.z-dn.net/?f=5x%2Bx%3D-20%2B8)
![6x=-12](https://tex.z-dn.net/?f=6x%3D-12)
![x=-2](https://tex.z-dn.net/?f=x%3D-2)
Now,
![8(0.5x-3)+4x=-9-(x+6)](https://tex.z-dn.net/?f=8%280.5x-3%29%2B4x%3D-9-%28x%2B6%29)
![4x-24+4x=-9-x-6](https://tex.z-dn.net/?f=4x-24%2B4x%3D-9-x-6)
![8x-24=-15-x](https://tex.z-dn.net/?f=8x-24%3D-15-x)
![8x+x=-15+24](https://tex.z-dn.net/?f=8x%2Bx%3D-15%2B24)
![9x=9](https://tex.z-dn.net/?f=9x%3D9)
![x=1](https://tex.z-dn.net/?f=x%3D1)
Therefore, the solutions of given equations
,
and
are x=-1, x=-2 and x=1 respectively.
It should be noted that some of the examples of situations of transformation where the image of the object appear different but represent the same thing include:
-
The rigid transformation of geometric and three-dimensional shapes.
- The transformation of bank notes to an electronic form of money.
- The transformation of liquid water to ice and steam.
- The transformation of cube sugar to granulated sugar.
<h3>Analyzing transformations.</h3>
It should be noted that algebraic operations can be performed on polynomials in order to give equivalent expressions that have the same value.
In the example given above, it can be deduced that even though, there is a transformation as the things appear different, but its meaning remains the same.
Learn more about transformation on:
brainly.com/question/26068179
Solution :
A). x = 2 (mod 3) ![$\mu = 3\times 5 \times 7 = 105$](https://tex.z-dn.net/?f=%24%5Cmu%20%3D%203%5Ctimes%205%20%5Ctimes%207%20%3D%20105%24)
x = 3 (mod 5) ![$y_1=35^{-1} (\mod 3)$](https://tex.z-dn.net/?f=%24y_1%3D35%5E%7B-1%7D%20%28%5Cmod%203%29%24)
x = 4 (mod 7) ![y_1=2](https://tex.z-dn.net/?f=y_1%3D2)
![$y_2=21^{-1}(\mod5) = 1$](https://tex.z-dn.net/?f=%24y_2%3D21%5E%7B-1%7D%28%5Cmod5%29%20%3D%201%24)
![$y_3=15^{-1}(\mod7) = 1$](https://tex.z-dn.net/?f=%24y_3%3D15%5E%7B-1%7D%28%5Cmod7%29%20%3D%201%24)
![$x=2 \times 35 \times 2 + 3\times 21\times 1+4\times 15\times 1$](https://tex.z-dn.net/?f=%24x%3D2%20%5Ctimes%2035%20%5Ctimes%202%20%2B%203%5Ctimes%2021%5Ctimes%201%2B4%5Ctimes%2015%5Ctimes%201%24)
![=140+63+60](https://tex.z-dn.net/?f=%3D140%2B63%2B60)
![=263](https://tex.z-dn.net/?f=%3D263)
≡ 53(mod 105)
Hence the solution is 105 k + 53 > 1000 for k = 10
Therefore, the minimum number of students = 1103
B). ![$\phi (935) = 640$](https://tex.z-dn.net/?f=%24%5Cphi%20%28935%29%20%3D%20640%24)
By Eulu's theory
if n and 935 are coprime.
Now,
and 80 x 8 = 640
⇒ g(n,935) = 1
⇒ 5, 11, 17 do not divide n