7\left(-\frac{7}{6}\right) + 3\left(\frac{21}{2}\right) + c = -4

["Understanding and Simplifying the Equation: 7 × (−7/6) + 3 × (21/2) + c = –4", "In algebra, solving equations step-by-step helps clarify complex expressions and builds strong problem-solving skills. Today, we’ll explore and simplify the equation:", "$$\n7 \left( -\frac{7}{6} \right) + 3 \left( \frac{21}{2} \right) + c = -4\n$$", "---", "### Step 1: Simplify Each Expression Separately", "We begin by simplifying each term in the equation:", "1. Simplify $ 7 \left( -\frac{7}{6} \right) $:\n Multiply numerator:\n $$\n 7 \ imes -\frac{7}{6} = -\frac{49}{6}\n $$", "2. Simplify $ 3 \left( \frac{21}{2} \right) $:\n Multiply numerator:\n $$\n 3 \ imes \frac{21}{2} = \frac{63}{2}\n $$", "---", "### Step 2: Rewrite the Equation", "Substitute the simplified values back into the equation:", "$$\n-\frac{49}{6} + \frac{63}{2} + c = -4\n$$", "---", "### Step 3: Find a Common Denominator", "To add or combine fractions, find a common denominator. The least common denominator between 6 and 2 is 6:", "Convert $ \frac{63}{2} $ to sixths:\n$$\n\frac{63}{2} = \frac{63 \ imes 3}{2 \ imes 3} = \frac{189}{6}\n$$", "Now rewrite the equation with a common denominator:", "$$\n-\frac{49}{6} + \frac{189}{6} + c = -4\n$$", "---", "### Step 4: Combine the Fractions", "Add the fractions:", "$$\n\left( -\frac{49}{6} + \frac{189}{6} \right) + c = -4\n\Rightarrow \frac{140}{6} + c = -4\n$$", "Simplify $ \frac{140}{6} $:", "$$\n\frac{140}{6} = \frac{70}{3}\n$$", "So the equation becomes:", "$$\n\frac{70}{3} + c = -4\n$$", "---", "### Step 5: Isolate Variable $ c $", "Subtract $ \frac{70}{3} $ from both sides:", "$$\nc = -4 - \frac{70}{3}\n$$", "Express $-4$ as a fraction with denominator 3:", "$$\n-4 = -\frac{12}{3}\n$$", "Now compute:", "$$\nc = -\frac{12}{3} - \frac{70}{3} = -\frac{82}{3}\n$$", "---", "### Final Answer", "$$\nc = -\frac{82}{3}\n$$", "---", "### Why This Equation Matters", "This problem reinforces core algebraic skills including:", "- Multiplying fractions\n- Simplifying expressions with negative numbers\n- Working with like denominators\n- Solving for an unknown variable", "Mastering such equations builds a solid foundation for more advanced topics in math, including calculus and linear algebra.", "If you're working through similar equations, practice simplifying step-by-step—this approach builds clarity and confidence!", "---", "Keywords: algebra worksheet, solve equations, fractional arithmetic, simplify expressions, solving for c, algebra simplification, linear equations, math problem-solving, combine like terms, fraction operations, algebraic expressions."]