y = \frac{3}{4} \cdot \left( -\frac{24}{25} \right) + 2 = -\frac{18}{25} + 2 = \frac{32}{25}
["# Simplifying a Linear Expression: A Step-by-Step Breakdown of ( y = \frac{3}{4} \cdot \left( -\frac{24}{25} \right) + 2 = -\frac{18}{25} + 2 = \frac{32}{25} )", "Solving algebraic expressions step by step is a fundamental skill in mathematics, whether you're simplifying equations, understanding function behavior, or preparing for more complex problems. One common operation involves multiplying fractions, adding constants, and combining terms — all of which are illustrated in the expression:", "[\ny = \frac{3}{4} \cdot \left( -\frac{24}{25} \right) + 2\n]", "In this article, we’ll carefully walk through simplifying the right-hand side of this equation to arrive at the clean result ( y = \frac{32}{25} ), explaining each step clearly for better comprehension.", "---", "### Step 1: Multiply the Fractions", "The first operation in the expression is multiplication:", "[\n\frac{3}{4} \cdot \left( -\frac{24}{25} \right)\n]", "When multiplying two fractions, multiply the numerators together and the denominators together:", "[\n\frac{3 \cdot (-24)}{4 \cdot 25} = \frac{-72}{100}\n]", "This simplifies further. We note that (-72/100) can be reduced, but we’ll keep the negative sign for clarity until final evaluation.", "---", "### Step 2: Perform the Division Equivalent via Multiplication", "Alternatively and often more efficiently, we can express the multiplication as division:", "[\n\frac{3}{4} \cdot \left( -\frac{24}{25} \right) = -\frac{3}{4} \cdot \frac{24}{25}\n]", "Now multiply numerators and denominators:", "[\n= -\frac{3 \cdot 24}{4 \cdot 25} = -\frac{72}{100}\n]", "Again, we obtain (-0.72) or (-\frac{72}{100}). This fractional form is preferred for exact calculations, especially before combining with other terms.", "---", "### Step 3: Rewrite Constant for Common Denominator", "To combine the multiplied fraction with the next term (+2), we express 2 as a fraction with the same denominator as (-\frac{72}{100}):", "[\n2 = \frac{2 \cdot 100}{100} = \frac{200}{100}\n]", "---", "### Step 4: Add the Two Fractions", "Now add:", "[\n-\frac{72}{100} + \frac{200}{100} = \frac{200 - 72}{100} = \frac{128}{100}\n]", "Wait — here we notice a slight inconsistency. Earlier, direct multiplication yielded (-72/100), and adding 2 gives (-72/100 + 200/100 = 128/100), but this contradicts the expected final value of (32/25). Let’s recheck carefully.", "---", "### Correcting and Verifying the Final Result", "Start again:", "[\ny = \frac{3}{4} \cdot \left( -\frac{24}{25} \right) + 2\n]", "First multiply:", "[\n\frac{3}{4} \cdot \left( -\frac{24}{25} \right) = -\frac{3 \cdot 24}{4 \cdot 25} = -\frac{72}{100} = -\frac{18}{25}\n]", "Now add 2:", "[\ny = -\frac{18}{25} + 2 = -\frac{18}{25} + \frac{50}{25} = \frac{50 - 18}{25} = \frac{32}{25}\n]", "---", "### Final Result", "Thus, the full simplification confirms:", "[\ny = \frac{3}{4} \cdot \left( -\frac{24}{25} \right) + 2 = -\frac{18}{25} + 2 = \frac{32}{25}\n]", "The decimal equivalent of ( \frac{32}{25} ) is (1.28), illustrating how fractions accurately represent this value.", "---", "### Why This Simplification Matters", "Understanding how to simplify expressions like this reinforces key algebraic skills:\n- Multiplying fractions is foundational for solving equations.\n- Properly combining constants with fractional terms ensures precision in calculations.\n- Step-by-step arithmetic prevents error and enhances problem-solving clarity.", "---", "### Summary", "Simplifying expressions step-by-step ensures accuracy and understanding. For the equation above:", "[\ny = \frac{3}{4} \cdot \left( -\frac{24}{25} \right) + 2 = \frac{32}{25}\n]", "we:\n- Multiply fractions: ( \frac{3 \cdot (-24)}{4 \cdot 25} = -\frac{18}{25} )\n- Express (2) as ( \frac{50}{25} )\n- Add: ( -\frac{18}{25} + \frac{50}{25} = \frac{32}{25} )", "This process demonstrates both computational strategy and algebraic rigor, empowering learners to confidently tackle more advanced mathematical challenges.", "---", "### Key Takeaways:", "- Break expressions into logical steps.\n- Multiply fractions by multiplying numerators and denominators.\n- Convert whole numbers to fractions with common denominators before adding.\n- Always simplify fully and verify final results.", "Mastering these steps not only solves this equation but builds a strong base for future learning in algebra, calculus, and beyond."]
