To find the value of $a$ that makes the equation $8x - 8 + 3ax = 5ax - 2a$ an identity, we need to follow these steps:

Step 1: Collect like terms.

Group the terms with $x$ on one side and the constant terms on the other side.

$8x + 3ax - 5ax = -2a + 8$

Step 2: Combine like terms.

Simplify the terms involving $x$:

$(8 + 3a - 5a)x = -2a + 8$

This simplifies to:

$(8 - 2a)x = -2a + 8$

Step 3: Set the coefficients equal.

For this equation to be an identity, the coefficient of $x$ and the constant term must both be 0. This is because an identity must hold true for all values of $x$. Therefore, we set both expressions equal to 0:

1. $8 - 2a = 0$
2. $-2a + 8 = 0$

Step 4: Solve for $a$.

From the first equation:

$8 - 2a = 0$ $2a = 8$ $a = 4$

Now, check if this value satisfies the second equation:

$-2(4) + 8 = 0$ $-8 + 8 = 0$

Since both conditions are satisfied, the value of $a$ that makes the equation an identity is $\boxed{4}$.

Would you like any more details or have any questions about this solution?

Related Questions:

1. What is an identity in algebraic terms?
2. How do you determine if an equation is an identity?
3. Can there be more than one value of $a$ that satisfies the condition of an identity?
4. How would the solution change if the equation involved quadratic terms?
5. What would happen if you had more than one variable in the equation?

Tip:

Always verify your solution by substituting the value back into the original equation to ensure both sides are equal.