6(-\frac{5}{3}) + b = 5 \Rightarrow -10 + b = 5 \Rightarrow b = 15
["Understanding the Equation: How to Solve for ( b ) in ( 6\left(-\frac{5}{3}\right) + b = 5 )", "When solving linear equations, one common challenge students and learners face is correctly simplifying and solving for a variable. A typical problem many encounter begins as:", "[\n6\left(-\frac{5}{3}\right) + b = 5\n]", "This equation may seem straightforward, but proper step-by-step handling is essential to arrive at the correct solution. Let’s break down the process step-by-step to clarify how we arrive at ( b = 15 ).", "---", "### Step 1: Simplify the Left Side", "Start by simplifying the multiplication on the left-hand side:", "[\n6 \ imes \left(-\frac{5}{3}\right) = -\frac{30}{3} = -10\n]", "Now, substitute this back into the equation:", "[\n-10 + b = 5\n]", "---", "### Step 2: Isolate the Variable ( b )", "To solve for ( b ), subtract ( -10 ) from both sides. Remember that subtracting a negative is the same as adding a positive:", "[\n-10 + b + 10 = 5 + 10\n]", "Simplify both sides:", "[\nb = 15\n]", "---", "### Step 3: Final Answer", "Thus, the solution to the equation ( 6\left(-\frac{5}{3}\right) + b = 5 ) is:", "[\n\boxed{b = 15}\n]", "---", "### Why This Matters for Learning Algebra", "Understanding how to correctly simplify expressions and isolate variables strengthens foundational algebra skills. This problem illustrates the importance of:", "- Performing arithmetic correctly, especially with fractions and negative numbers.\n- Maintaining balance when performing operations on both sides of an equation.\n- Isolating the variable to find its value clearly.", "Mastering these steps enables learners to tackle more complex equations with confidence.", "---", "Key Takeaway:\nStart by simplifying multiplication carefully, then move step-by-step to isolate the variable—this method consistently leads to accurate results in linear equations.", "If you’re studying algebra, practice with similar equations to build fluency—remember: every operation affects the whole equation, so clarity and care are key!"]
