Solving, \( 4 + x = 0.6(10 + x) \) leads to \( 4 + x = 6 + 0.6x \).

["# Solving ( 4 + x = 0.6(10 + x) ): A Step-by-Step Guide to Understanding the Process", "Solving linear equations is a fundamental skill in algebra, essential for mastering more advanced math topics. One classic example is solving the equation:\n[ 4 + x = 0.6(10 + x) ]\nThis equation may appear simple, but it reveals key algebraic principles when solved step-by-step — especially understanding how distributive properties and combining like terms play a role. In this article, we’ll walk through the full solution and explain how it leads naturally to the equivalent form:\n[ 4 + x = 6 + 0.6x ]", "## What Is the Equation Trying to Express?", "Before solving, it’s helpful to understand what the equation means. On the left, ( 4 + x ) represents a quantity — the sum of 4 and an unknown value ( x ). On the right, ( 0.6(10 + x) ) means 60% of a total that includes 10 plus ( x ). The equality states that these two expressions are equal — a balance of values.", "## Step 1: Expand the Right Side Using the Distributive Property", "To simplify, we apply the distributive property, a crucial rule in algebra:\n[ a(b + c) = ab + ac ]\nApply this to the right-hand side:\n[ 0.6(10 + x) = 0.6 \ imes 10 + 0.6 \ imes x = 6 + 0.6x ]", "Now the equation becomes:\n[ 4 + x = 6 + 0.6x ]\nThis is the simplified form we were asked to derive.", "## Step 2: Rearranging the Equation to Collect Like Terms", "Now that both sides are expanded, we proceed to isolate ( x ) using algebraic operations:", "### Subtract ( 0.6x ) from Both Sides\nThis cancels ( x ) on the right and moves it to the left:\n[ 4 + x - 0.6x = 6 + 0.6x - 0.6x ]\n[ 4 + 0.4x = 6 ]", "### Subtract 4 from Both Sides\nNow isolate the term with ( x ):\n[ 4 + 0.4x - 4 = 6 - 4 ]\n[ 0.4x = 2 ]", "## Step 3: Solve for ( x )", "Now divide both sides by 0.4:\n[ x = \frac{2}{0.4} = 5 ]", "## Why This Works: The Logic Behind the Steps", "Every operation preserves the equality, ensuring the equation remains balanced. Distributing removes parentheses, turning expressions into easier-to-manage forms. Subtracting and dividing flexibly adjust terms — a core skill in algebra.", "From ( 4 + x = 0.6(10 + x) ), we transformed the notation, but the solution ( x = 5 ) holds regardless of format. This demonstrates how algebraic equivalence allows different representations while preserving meaning.", "## Real-World Applications", "Equations like this appear in budgeting, science, and engineering — anywhere values depend linearly on variables. For example, calculating fixed and variable costs often involves expressions like ( a + bx ), balancing inputs and rates.", "## Conclusion", "Solving ( 4 + x = 0.6(10 + x) ) step-by-step illustrates vital algebraic techniques: distribution, combining like terms, and isolating variables. Through these, we transform the original equation into the equivalent and solved form:\n[ 4 + x = 6 + 0.6x ]\nMastering these steps builds a strong foundation for algebra, preparing you to tackle more complex equations with confidence.", "If you're refining your algebraic skills, start practicing such equations — transforming and simplifying them is key to fluency. Keep solving, keep learning!"]