To solve the problem in the image, we need to carefully analyze the given vector, translation details, and equations implied in the description. Let's break it down step by step:

Problem Details:

1. Translation Vector: $\langle 4, 1 \rangle$

   • This vector represents the movement in the $x$- and $y$-directions during the translation.

2. Points Involved:

   • $A(-1, w) \to A'(2x + 1, 4)$
   • $B(8y - 1, 1) \to B'(3, 3z)$

3. Objective:

   • Find the values of $w$, $x$, $y$, and $z$.

Translation Relation:

The vector $\langle 4, 1 \rangle$ means:

• The $x$-coordinate of any point increases by $4$.
• The $y$-coordinate of any point increases by $1$.

This gives us:

1. $A' = A + \langle 4, 1 \rangle$
2. $B' = B + \langle 4, 1 \rangle$

Equations:

For Point $A$:

From $A(-1, w)$ to $A'(2x + 1, 4)$:

• $x$-coordinate: $2x + 1 = -1 + 4$
• $y$-coordinate: $4 = w + 1$

For Point $B$:

From $B(8y - 1, 1)$ to $B'(3, 3z)$:

• $x$-coordinate: $3 = (8y - 1) + 4$
• $y$-coordinate: $3z = 1 + 1$

Solve for Variables:

1. From $2x + 1 = -1 + 4$: $2x + 1 = 3 \implies 2x = 2 \implies x = 1$

2. From $4 = w + 1$: $w = 4 - 1 \implies w = 3$

3. From $3 = (8y - 1) + 4$: $3 = 8y + 3 \implies 8y = 0 \implies y = 0$

4. From $3z = 1 + 1$: $3z = 2 \implies z = \frac{2}{3}$

Final Answer:

$w = 3, \, x = 1, \, y = 0, \, z = \frac{2}{3}$

Let me know if you would like more explanation or if you have further questions!

Expanding Understanding:

Here are 5 related questions for practice:

1. How do you find a translation vector given two points before and after translation?
2. How would the equations change if the translation vector was $\langle -4, 3 \rangle$?
3. Can you derive the general formula for translations in 3D space?
4. What happens if the coordinates include negative values in the vector?
5. How can you verify the translation is correct for both points?

Tip:

Always write down the translation vector explicitly and apply it systematically to avoid mistakes with signs.