Problem 7 - Introductory Algebra for College Students

Question

Solve each equation. Be sure to check your proposed solution by substituting it for the variable in the original equation. $$4(x+1)=20$$

Step-by-Step Solution

Answer

The solution to the equation \(4(x + 1) = 20\) is \(x = 4\).

Step 1: Distribute the 4 to the Inside of the Parentheses

Apply the distributive property to remove the parentheses: \(4*(x+1) = 4*x + 4*1\). This simplifies to: \(4x + 4 = 20\).

Step 2: Isolate the variable term

In order to get x alone on one side of the equation, subtract 4 from both sides: \(4x + 4 - 4 = 20 - 4\). This simplifies to: \(4x = 16\)

Step 3: Solve for x

Divide both sides of the equation by 4 to solve for x: \(4x/4 = 16/4\). This simplifies to: \(x = 4\)

Step 4: Check the solution

Substitute x = 4 into the original equation: \(4(4 + 1) = 20\). Simplifying gives: \(20 = 20\). The equation balances, so x = 4 is the correct solution

Key Concepts

Distributive Property

The distributive property is an essential tool in simplifying expressions, particularly when dealing with equations that have parentheses. This property allows us to multiply a single term by each term inside the parentheses. In the original exercise, we had to apply the distributive property to the equation \(4(x+1) = 20\).

Here's how it works:

• Multiply the number outside the parentheses (4) by each term inside the parentheses \(x+1\).
• This results into: \(4 \cdot x + 4 \cdot 1\).
• Simplifying this, we get \(4x + 4\).

By using the distributive property, we've turned an equation with parentheses into one that's easier to manage. This step is crucial because it sets the stage for the subsequent steps where we'll isolate the variable to find the solution.

Isolate the Variable

To solve a linear equation like \(4x + 4 = 20\), we need to isolate the variable, \(x\), on one side of the equation. Isolating the variable means getting \(x\) alone on one side, which helps in clearly identifying the value of \(x\).

Let's break it down further:

• Initially, you have the equation \(4x + 4 = 20\).
• Subtract 4 from both sides to eliminate the constant term on the left side: \(4x + 4 - 4 = 20 - 4\).
• This simplifies to \(4x = 16\).

At this point, the variable term is isolated. The next step involves dividing both sides by the coefficient of \(x\) which is 4, leading to \(x = 4\). This process of isolating the variable transforms a complicated equation into a simple one where you can easily identify the value of \(x\).

Substitute to Verify Solution

Substituting the solution back into the original equation is a vital step to verify whether the solution is correct. This step assures us that the value obtained for \(x\) actually satisfies the given equation.

Here's how it works in practice:

• Our proposed solution to the equation \(4(x+1)=20\) was \(x = 4\).
• Substitute \(x\) with 4: \(4(4 + 1)\).
• Calculate inside the parentheses first: \(4 + 1 = 5\).
• Multiply: \(4 \times 5 = 20\).

This shows that substituting \(x = 4\) results in both sides of the equation equaling 20, proving that our solution \(x = 4\) is indeed correct. Substitution not only confirms the accuracy of our solution but also strengthens our confidence in the process we followed to solve the equation.