Problem 47 - Algebra A Combined Function

Question

Solve each equation. See Examples 9 and \(10 .\) \(-4 x+20=4 x-20\)

Step-by-Step Solution

Answer

The solution to the equation is \(x = 5\).

Step 1: Eliminate One Variable Term

To eliminate the variable term on one side, add \(4x\) to both sides of the equation:\[-4x + 4x + 20 = 4x + 4x - 20\]This simplifies to:\[20 = 8x - 20\]

Step 2: Isolate the Variable Term

Add 20 to both sides to get the variable term by itself:\[20 + 20 = 8x - 20 + 20\]This simplifies to:\[40 = 8x\]

Step 3: Solve for the Variable

Divide both sides by 8 to solve for \(x\):\[\frac{40}{8} = \frac{8x}{8}\]This simplifies to:\[x = 5\]

Key Concepts

Eliminating Variable Terms

When tackling linear equations, eliminating variable terms from one side of the equation is often the first step. This means we want to remove any instances of the unknown variable (like \(x\)) from one side to simplify the equation. In our example, we have

• \(-4x + 20 = 4x - 20\)

To eliminate the \(-4x\) from the left side, we can add \(4x\) to both sides of the equation. This is because what we do to one side we must do to the other to maintain balance. Once we perform the addition, the equation transforms to:

• \( -4x + 4x + 20 = 4x + 4x - 20 \)

which simplifies to:

• \(20 = 8x - 20\)

At this point, the variable \(x\) is only present on one side of the equation, making it easier for further isolating.

Isolating Variable Terms

With the variable only on one side post-elimination, our next task is isolating it. Isolation involves getting the variable by itself on one side of the equation. In other words, we aim to transform

• \(20 = 8x - 20\)

We begin by eliminating any constants that accompany the variable. In this instance, we remove \(-20\) by adding \(20\) to both sides:

• \(20 + 20 = 8x - 20 + 20\)

This results in:

• \(40 = 8x\)

Now, \(8x\) stands alone, making the next step clearer. This transformation is crucial as it fully isolates the variable term from any constant terms, preparing to solve for the variable itself.

Step-by-Step Solutions

The beauty of step-by-step solutions lies in their clear, sequential nature, guiding us through each transformation of the equation. By systematically applying the fundamental principles of algebra—like keeping equations balanced and performing operations equally on both sides—solving becomes a straightforward task. Let's revisit the key stages from our equation:

• First, we eliminated the variable term \(-4x\) by adding \(4x\) to each side.
• Next, we isolated \(8x\) by adding \(20\) to both sides.
• Finally, solving for \(x\) entailed dividing each side by \(8\) to find \(x = 5\).

These steps break down the problem into manageable parts, highlighting each change and maintaining the equation's balance. This structured approach builds confidence and ensures that every action is justified, helping students to understand not just how, but why each step is necessary in the problem-solving process.