Understanding the Context

Why Solving Linear Equations Matters

Linear equations such as 8x – 4 = 5x + 15 appear frequently in classrooms, standardized tests, and real-world applications like budgeting, physics, and computer science. Knowing how to isolate variables and simplify expressions is essential for success in STEM fields.

Step-by-Step Solution to 8x – 4 = 5x + 15

Key Insights

Step 1: Understand the Equation

We start with:
8x – 4 = 5x + 15
Our goal is to find the value of x that makes both sides equal.

Step 2: Move Variables to One Side

To isolate x, subtract 5x from both sides to eliminate it from the right:
8x – 5x – 4 = 5x – 5x + 15
Which simplifies to:
3x – 4 = 15

Step 3: Eliminate the Constant

Next, add 4 to both sides to move the constant to the right:
3x – 4 + 4 = 15 + 4
This gives:
3x = 19

Step 4: Solve for x

Finally, divide both sides by 3:
3x ÷ 3 = 19 ÷ 3
x = 19/3
or
x ≈ 6.33 (in decimal form)

Final Thoughts

Final Answer

x = 19/3 (exact form)
or approximately x = 6.33

Real-World Applications of Linear Equations

Understanding how to solve equations like 8x – 4 = 5x + 15 helps model real-life scenarios:

• Finance: Comparing loan payments or comparing costs over time.
  
• Science: Calculating rates, such as speed or chemical concentration.
  
• Technology: Debugging or optimizing algorithms that rely on linear relationships.

Tips to Master Solving Linear Equations

1. Always perform the same operation on both sides to keep the equation balanced.
   
2. Keep expressions simplified after each step.
   
3. Use parentheses to avoid algebraic errors.
   
4. Practice regularly with varied numbers and variable arrangements.
   
5. Double-check solutions by substituting x = 19/3 back into the original equation.