To find the value of \( a \) in the equation \( 8x - 8 + 3a = 5ax - 2a \), follow these steps:

### Step 1: Simplify and group like terms
We want to gather terms with \( x \) and terms without \( x \) on each side of the equation.

\[
8x - 8 + 3a = 5ax - 2a
\]

Group the terms involving \( x \) together and the constants together:

\[
8x - 5ax = 2a - 3a + 8
\]

### Step 2: Simplify further
Now, simplify both sides:

\[
(8 - 5a)x = (2a - 3a + 8)
\]

This simplifies to:

\[
(8 - 5a)x = -a + 8
\]

### Step 3: Analyze the result
Since the equation must hold for all \( x \), the coefficients of \( x \) must be equal, and the constants must be equal as well. Therefore, we set up two equations:

1. \( 8 - 5a = 0 \) (for the \( x \)-terms)
2. \( -a + 8 = 0 \) (for the constant terms)

### Step 4: Solve for \( a \)

#### Equation 1: \( 8 - 5a = 0 \)
\[
5a = 8
\]
\[
a = \frac{8}{5}
\]

#### Equation 2: \( -a + 8 = 0 \)
\[
a = 8
\]

### Step 5: Conclusion
Since we obtained two different values for \( a \), there is a contradiction. This means that there is **no value of \( a \)** that satisfies the given equation for all \( x \).

Would you like to discuss any specific part of this solution? Here are some related questions to expand on:

1. How do we handle contradictions in linear equations?
2. What does it mean if there is no solution to an equation?
3. How can you check if an equation has infinite solutions?
4. What are the conditions for a system of equations to have a unique solution?
5. Can we solve this equation for specific values of \( x \) instead?

**Tip:** When solving equations, always check for consistency to determine if there are solutions or contradictions.

Explore the step-by-step solution to find the value of a in the equation 8x - 8 + 3a = 5ax - 2a. This problem covers key algebraic concepts, such as equating coefficients, dealing with contradictions, and analyzing linear equations. Ideal for students in Grades 8-10.

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