To find the coordinates of the point $A'$ after translating the point $A(-4, 3)$ by the vector $\langle 5, -1 \rangle$, you can use the following steps:

1. Identify the original coordinates of $A$: $A = (-4, 3)$

2. Identify the components of the translation vector $\langle 5, -1 \rangle$: $\text{Vector} = (5, -1)$

3. Apply the translation by adding each component of the vector to the corresponding coordinate of $A$: $A'_x = -4 + 5 = 1$ $A'_y = 3 + (-1) = 2$

4. Thus, the coordinates of $A'$ are: $A' = (1, 2)$

Answer:

The coordinates of $A'$ are $(1, 2)$.

Would you like further explanation on this solution, or do you have any additional questions?

Here are some related questions that could expand on this information:

1. How do you determine the translation vector for different transformations?
2. What would be the coordinates of $A'$ if the vector were $\langle -5, 1 \rangle$ instead?
3. How would this solution change if we were performing a reflection instead of a translation?
4. What is the general rule for translating a point $(x, y)$ by a vector $\langle a, b \rangle$?
5. How would you graph the original and translated points on a coordinate plane?

Tip: When translating points, add the components of the vector to the corresponding coordinates of the point to get the new position easily.