We are given the equation of a curve $y = ax + b$, and we know it passes through the points $(2, 0)$ and $(4, 8)$. This allows us to set up two equations based on these points:

1. When $x = 2$ and $y = 0$: $0 = 2a + b$ This simplifies to: 2a + b = 0 \tag{1}

2. When $x = 4$ and $y = 8$: $8 = 4a + b$ This simplifies to: 4a + b = 8 \tag{2}

Now, let's solve this system of equations.

Step 1: Solve for $b$ in equation (1)

From equation (1), we can express $b$ in terms of $a$: $b = -2a$

Step 2: Substitute $b$ into equation (2)

Substituting $b = -2a$ into equation (2): $4a + (-2a) = 8$ $2a = 8$ $a = 4$

Step 3: Find $b$

Now substitute $a = 4$ into $b = -2a$: $b = -2(4) = -8$

Thus, the values of $a$ and $b$ are: $a = 4, \quad b = -8$

The equation of the curve is $y = 4x - 8$.

Would you like further details or have any other questions?

Here are 5 related questions to explore further:

1. What is the slope of the line $y = 4x - 8$?
2. How would the equation change if the curve passed through another point?
3. What are the x- and y-intercepts of the equation $y = 4x - 8$?
4. How can you graph the line $y = 4x - 8$ using the points you have?
5. How do you solve a system of equations using substitution or elimination?

Tip: Always check your solutions by substituting them back into the original equations to ensure accuracy.