A translation moves point $A(-3,2)$ to the right 5 units and up 3 units. This translation is done a total of 6 times. After these translations, the point is at $(x, y)$. What is the value of $x+y$ ?
(A) 34
(B) 49
(C) 53
(D) 47
(E) 43

![](https://cdn.mathpix.com/cropped/2024_04_20_6027bc27089ed4fc493cg-024.jpg?height=341&width=399&top_left_y=1521&top_left_x=1302)
Solution 1

Each translation to the right 5 units increases the $x$-coordinate of point $A$ by 5 .

Similarly, each translation up 3 units increases the $y$-coordinate of point $A$ by 3 .

After 1 translation, the point $A(-3,2)$ would be at $B(-3+5,2+3)$ or $B(2,5)$.

After 2 translations, the point $A(-3,2)$ would be at $C(2+5,5+3)$ or $C(7,8)$.

After 3 translations, the point $A(-3,2)$ would be at $D(7+5,8+3)$ or $D(12,11)$.

After 4 translations, the point $A(-3,2)$ would be at $E(12+5,11+3)$ or $E(17,14)$. After 5 translations, the point $A(-3,2)$ would be at $F(17+5,14+3)$ or $F(22,17)$. After 6 translations, the point $A(-3,2)$ would be at $G(22+5,17+3)$ or $G(27,20)$. After these 6 translations, the point is at $(27,20)$ and so $x+y=27+20=47$.

## Solution 2

Each translation to the right 5 units increases the $x$-coordinate of point $A$ by 5 .

Similarly, each translation up 3 units increases the $y$-coordinate of point $A$ by 3 .

Therefore, each translation of point $A(-3,2)$ to the right 5 units and up 3 units increases the sum of the $x$ - and $y$-coordinates, $x+y$, by $5+3=8$.

After 6 of these translations, the sum $x+y$ will increase by $6 \times 8=48$.

The sum of the $x$ - and $y$-coordinates of point $A(-3,2)$ is $-3+2=-1$.

After these 6 translations, the value of $x+y$ is $-1+48=47$.

ANSWER: (D)

#