Question

Solve the equation. $$ 2(v+1)=4 $$

Short answer

The solution for the equation is \(v = 1\).

Step-by-step solution

Distribute the term on the left

Multiply 2 with each term inside the brackets. The equation becomes \(2v + 2 = 4\).

Subtract 2 from both sides

To find 'v' by itself, subtract 2 from both sides of the equation, getting \(2v = 4 - 2\). The equation now becomes \(2v = 2\).

Solve variable v

To solve for variable 'v', divide each side of the equation by 2. The final answer is \(v = 1\).

Key concepts

The Distributive Property

Understanding the distributive property is vital in simplifying algebraic expressions and solving equations. Essentially, it allows you to multiply a single term by terms within parentheses. For instance, in the equation from our exercise, we have 2(v+1)=4. The distributive property is applied by multiplying 2 with each term inside the parentheses, which yields 2v + 2 = 4.

This property provides a method for removing parentheses which is frequently the first step in simplifying equations. It follows the formula: a(b + c) = ab + ac. By applying this, we distribute the multiplication of 'a' over the addition inside the parentheses, resulting in two separate terms.

Isolating Variables

To isolate the variable means to get the variable on one side of the equation by itself, which is the ultimate goal when trying to solve linear equations. It involves performing operations that reverse what has been done to the variable. In our example, after applying the distributive property, we end up with 2v + 2 = 4. To isolate 'v', we need to undo the addition of 2 and the multiplication by 2.

First, subtract 2 from both sides to cancel the '+2' on the side with the variable. This gives us 2v = 2. Next, we divide both sides by 2 to undo the multiplication. The result is v = 1. These 'inverse operations' are essential tools for isolating variables.

Equivalent Equations

An equivalent equation is one that has the same solution as the original equation, even if it looks different. During the process of solving an equation, we perform a series of operations that transform the original equation into simpler ones, all of which are equivalent to each other.

In our exercise, we start with 2(v+1)=4, apply the distributive property and get 2v + 2 = 4. We then subtract 2 from both sides resulting in 2v = 2. Finally, dividing each side by 2, we obtain v = 1. Each step produces an equivalent equation because each transformation is done by performing legal algebraic operations that maintain the equality—the balance—of both sides of the equation.

Understanding that these operations don't change the 'truth' of the equation helps students see that algebra is a process of reshaping equations to find the variable's value without altering that value.