The curve y = ax: + bx passes through (2, 0) and (4, 8).
Find a and b.

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.

The curve y = ax + bx passes through (2, 0) and (4, 8). Find a and b.

Learn how to solve for the values of a and b in the linear equation y = ax + b, given the points (2, 0) and (4, 8). This problem demonstrates the substitution method to solve a system of linear equations and is suitable for students in Grades 8-10.

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