Multiply second equation by 3: \( 12x - 3y = 30 \)

["### How to Multiply the Second Equation by 3: Simplifying Linear Equations (with (12x - 3y = 30))", "When working with linear equations in algebra, one common task is scaling equations to make solving systems of equations easier — and a practical example is multiplying the equation (12x - 3y = 30) by 3. In this article, we’ll walk through the process step-by-step and explain why this manipulation is helpful in solving equations and systems of equations.", "---", "#### What Does Multiplying an Equation by 3 Mean?", "Multiplying an entire equation by a constant—such as 3—does not change the solution set of the equation. It simply expresses the same relationship using a different coefficient, which can make substitution or elimination methods clearer when solving systems.", "Given the equation:\n[\n12x - 3y = 30\n]", "Multiplying both sides by 3 yields:\n[\n3 \cdot (12x - 3y) = 3 \cdot 30\n]\n[\n36x - 9y = 90\n]", "---", "#### Why Modify the Equation?", "- Easier Elimination: If you plan to eliminate a variable using elimination, working with more simplified coefficients (such as integers without common factors) can streamline the process.\n- Consistency in Coefficients: Matching numerical patterns between equations improves clarity when solving systems.\n- Standard Form Usage: Many methodologies assume equations in standard linear form, and scaling helps maintain uniformity.", "---", "#### A Quick Example: Solve Using Elimination", "Let’s see how multiplying by 3 helps:", "We now have:\n1. Original: (12x - 3y = 30)\n2. Scaled: (36x - 9y = 90)", "To eliminate one variable, notice that the (y)-coefficients in the two equations are multiples:\nEquation 1: (-3y), Equation 2: (-9y)", "Multiply Equation 1 by 3 to make coefficients match:\n(3 \cdot (12x - 3y) = 36x - 9y = 90)", "Now, subtract or manipulate equations confidently — for example, subtracting scaled versions to eliminate (y).", "---", "#### Final Simplified Form", "After multiplying the original equation by 3, the refined version becomes:\n[\n36x - 9y = 90\n]", "This form maintains equivalent meaning but often simplifies next algebraic steps.", "---", "#### Key Takeaways", "- Multiplying both sides of an equation by a non-zero constant preserves the equality and solution set.\n- Scaling equations is useful when solving systems via substitution or elimination.\n- In this case, multiplying (12x - 3y = 30) by 3 gives (36x - 9y = 90), offering a clearer or more convenient form for further manipulation.", "---", "Conclusion:\nMultiplying the equation (12x - 3y = 30) by 3 is a strategic step to simplify solving linear systems. While the meaning stays unchanged, this transformation enhances clarity and facilitates efficient elimination methods used widely in algebra. Understanding when and how to scale equations empowers stronger problem-solving skills in mathematics.", "---", "If you’re studying equations or systems of equations, remember: manipulating equations properly doesn’t change their solutions — but it can make finding those solutions much simpler. Multiplying by 3 is a basic but powerful tool in your algebra toolkit."]