Multiplica la segunda ecuación por 3: \( 12x - 3y = 15 \).

["Title: Simplify Your Algebra: Multiply the Second Equation by 3 in (12x - 3y = 15)", "When working with systems of equations, one common task is transforming equations to make solving easier. In this article, we’ll explore how multiplying the equation (12x - 3y = 15) by 3 affects its form and how it simplifies further algebraic manipulation.", "---", "### Understanding the Original Equation", "We start with:\n[\n12x - 3y = 15\n]\nThis is a linear equation in two variables, (x) and (y). It defines a straight line on the coordinate plane, and combining it with another equation helps solve systems using methods like substitution or elimination.", "---", "### Step 1: Multiply the Entire Equation by 3", "Multiplying both sides by 3 gives:\n[\n3 \cdot (12x - 3y) = 3 \cdot 15\n]\nDistribute the 3:\n[\n36x - 9y = 45\n]", "So, multiplying the equation (12x - 3y = 15) by 3 produces the new equation:\n[\n36x - 9y = 45\n]", "---", "### Why Multiply by 3?", "Though in this case it might seem unnecessary, multiplying by 3 can be strategically useful:", "- Align coefficients for elimination: If combined with another equation, like (12x - 3y = 15), multiplying helps make corresponding variables’ coefficients work together.\n- Clear fractions: Using fractions later becomes simpler with integer coefficients.\n- Standardize form: Having integer coefficients improves readability and reduces computational errors.", "---", "### Using the Transformed Equation in a System", "Let’s say we now have the system:\n[\n\begin{cases}\n12x - 3y = 15 \\n36x - 9y = 45\n\end{cases}\n]", "Notice both equations are now integer multiples—specifically, the second is just (3 \ imes) the first. This means they represent the same line, and the system has infinitely many solutions along the line (12x - 3y = 15).", "To find a unique solution, you’d typically use a different, non-multiple equation. But multiplying is key for consistent elimination methods.", "---", "### How to Solve the System", "Using elimination: subtract (3 \ imes (12x - 3y = 15)):\n[\n36x - 9y = 45\n]\n[\n36x - 9y = 45\n]\nSubtracting gives:\n[\n0 = 0 \quad \ ext{(a true but unhelpful statement)}\n]", "This confirms dependence — the equations are parallel and coincide.", "---", "### Final Thoughts", "Multiplying equations by scalars like 3 is a foundational tool in algebra. It preserves equality, helps streamline solving systems, and maintains clarity in coefficients. Remember: multiplying both sides by the same non-zero number keeps the solution set unchanged, making it a powerful step in advanced problem-solving.", "---", "Keywords:\nMultiply equation by 3, simplify algebra, linear equations system, solve (12x - 3y = 15), algebraic manipulation, elimination method, proportional equations.", "---", "Summary:\nMultiplying (12x - 3y = 15) by 3 transforms it to (36x - 9y = 45), aligning it mathematically for elimination but reinforcing the need to combine with independent equations for unique solutions. Mastering such steps boosts fluency in solving systems involving multiple equations."]