Solving Equations Using the Distributive Property

To solve this equation, follow these steps:

• Step 1: Apply the distributive property to the equation:

  $2(4-x) = 2 \times 4 - 2 \times x = 8 - 2x$

• Step 2: Simplify the equation:

  The equation now becomes: $8 - 2x = 8$

• Step 3: Isolate the variable $x$ by simplifying the equation:

  First, subtract 8 from both sides:
  $8 - 2x - 8 = 8 - 8$
  This simplifies to:
  $-2x = 0$

• Step 4: Solve for $x$ by dividing both sides by -2:

  $x = \frac{0}{-2} = 0$

Therefore, the solution to the equation is $x = 0$.

To open parentheses we will use the formula:

$a(x+b)=ax+ab$

$(7\times-2x)+(7\times5)=77$

We multiply accordingly

$-14x+35=77$

We will move the 35 to the right section and change the sign accordingly:

$-14x=77-35$

We solve the subtraction exercise on the right side and we will obtain:

$-14x=42$

We divide both sections by -14

$\frac{-14x}{-14}=\frac{42}{-14}$

$x=-3$

We open the parentheses according to the formula:

$a(x+b)=ax+ab$

$5\times x+5\times3=0$

$5x+15=0$

We will move the 15 to the right section and keep the corresponding sign:

$5x=-15$

Divide both sections by 5

$\frac{5x}{5}=\frac{-15}{5}$

$x=-3$

To solve the linear equation $8a + 2(3a - 7) = 0$, we'll proceed with the following steps:

Step 1: Apply the Distributive Property.
The equation given is $8a + 2(3a - 7) = 0$.
First, distribute the 2 across the terms inside the parenthesis:
$2(3a - 7) = 2 \times 3a + 2 \times (-7) = 6a - 14$.
By substituting this back into the equation, we have:
$8a + 6a - 14 = 0$.

Step 2: Combine Like Terms.
Now, combine the terms containing $a$:
$8a + 6a = 14a$.
The equation now becomes:
$14a - 14 = 0$.

Step 3: Isolate the Variable.
Add 14 to both sides of the equation to isolate terms with $a$:
$14a - 14 + 14 = 0 + 14$, which simplifies to:
$14a = 14$.
Next, divide both sides by 14 to solve for $a$:
$a = \frac{14}{14} = 1$.

Therefore, the solution to the equation is $a = 1$.

Let's proceed to solve the linear equation $3(a+1) - 3 = 0$:

Step 1: Distribute the 3 in the expression $3(a+1)$.

We get:
$3 \cdot a + 3 \cdot 1 - 3 = 0$

This simplifies to:
$3a + 3 - 3 = 0$

Step 2: Simplify the expression by combining like terms.

We simplify this to:
$3a + 0 = 0$ or simply $3a = 0$

Step 3: Isolate $a$ by dividing both sides by 3.

$\frac{3a}{3} = \frac{0}{3}$

Thus,
$a = 0$

Therefore, the solution to the problem is $a = 0$.

The correct choice is the option corresponding to $a = 0$.