Multiply the second equation by 3: \(12x - 3y = 27\).

["How to Multiply a Linear Equation by 3: Simplifying (12x - 3y = 27) Step by Step", "When working with equations in algebra, a common operation is multiplying both sides by a constant to simplify or restructure the equation. One practical application is multiplying the second equation in systems of equations by 3 to align coefficients. In this article, we’ll explore how to multiply the given equation (12x - 3y = 27) by 3, using it effectively in equation manipulation.", "---", "### Step-by-Step: Multiply (12x - 3y = 27) by 3", "Original equation:\n[\n12x - 3y = 27\n]", "Multiply every term by 3:\n[\n3 \cdot (12x) - 3 \cdot (3y) = 3 \cdot 27\n]", "Simplify each term:\n[\n36x - 9y = 81\n]", "Now, the equation (3(12x - 3y) = 3 \cdot 27) becomes:\n[\n36x - 9y = 81\n]\nThis is the equation obtained by multiplying the second equation by 3.", "---", "### Why Multiply Equations by Constants?", "Multiplying an equation by a non-zero constant preserves the equation’s solutions. It’s especially useful when solving systems of equations because it allows you to eliminate variables when combined with another equation. In this case, transforming (12x - 3y = 27) to (36x - 9y = 81) gives standardized coefficients that may align better with other equations in a system.", "---", "### How to Use This in Solving Systems of Equations", "Suppose you have the system:\n[\n\begin{cases}\n12x - 3y = 27 \\n3y = 4x + 9\n\end{cases}\n]", "Instead of working with (3y = 4x + 9), multiplying the first equation by 3 helps create compatible coefficients:\n[\n3 \cdot (12x - 3y) = 3 \cdot 27 \implies 36x - 9y = 81\n]\nRewriting the second equation with common structure:\n[\n3y = 4x + 9 \implies 9y = 12x + 27 \quad \ ext{(multiplying both sides by 3)}\n]", "Now, adding the two equations eliminates (y):\n[\n(36x - 9y) + (9y) = 81 + (12x + 27)\n]\n(Note: proper alignment may require rewriting carefully.)", "---", "### Summary", "- Multiplying (12x - 3y = 27) by 3 converts it to (36x - 9y = 81).\n- This preserves equivalence but adjusts coefficients for easier combination with other equations.\n- Such operations streamline solving systems of linear equations by standardizing variables’ coefficients.\n- This technique supports efficient algebraic manipulation and is foundational in linear algebra.", "---", "Conclusion:\nUnderstanding how to multiply equations, such as scaling (12x - 3y = 27) by 3, is a key skill for simplifying equations and solving systems efficiently. Whether you're studying algebra, preparing for calculus, or building problem-solving confidence, mastering equation multiplication unlocks faster and clearer solutions.", "---", "Keywords: multiply equation by 3, algebra simplification, solve linear equations, system of equations, equation manipulation, linear algebra, algebraic operations\nMeta description: Learn how to multiply the second equation (12x - 3y = 27) by 3, step-by-step. Understand significance in systems of equations and algebraic problem-solving."]