Solving the Algebraic Equation

The question asks us to find the value of the variable x in the given linear equation:

$$5x + 5(2 - x) + 10 = 4x - 6$$

We need to simplify this equation step-by-step to isolate x.

Simplifying the Expression

First, let's distribute the 5 across the terms inside the parentheses on the left side of the equation:

$$5x + (5 \times 2) - (5 \times x) + 10 = 4x - 6$$

$$5x + 10 - 5x + 10 = 4x - 6$$

Now, combine the like terms on the left side. The terms involving x are 5x and -5x, and the constant terms are 10 and 10.

$$ (5x - 5x) + (10 + 10) = 4x - 6 $$

$$ 0x + 20 = 4x - 6 $$

$$ 20 = 4x - 6 $$

Calculating the Value of x

Next, we want to get all the terms involving x on one side and the constant terms on the other side. Let's add 6 to both sides of the equation:

$$ 20 + 6 = 4x - 6 + 6 $$

$$ 26 = 4x $$

Finally, to find the value of x, divide both sides by 4:

$$ \frac{26}{4} = \frac{4x}{4} $$

$$ x = \frac{26}{4} $$

Performing the division:

$$ x = 6.5 $$

Conclusion

The calculated value of x is 6.5. This matches the third option provided.