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

["Multiply the Second Equation by 3: A Step Toward Solving Linear Systems", "When solving systems of linear equations, one essential step is simplifying and manipulating equations to make them easier to work with. In this article, we’ll explore what it means to multiply the second equation by 3—specifically, how this transformation enhances equation-solving strategies, particularly in context with the equation (12x - 3y = 30).", "---", "### Understanding the Original Equation", "The equation (12x - 3y = 30) is a linear equation in two variables, (x) and (y). It represents a straight line on the coordinate plane and is commonly used in algebra, economics, geometry, and many applied fields.", "To work toward finding values of (x) and (y) that satisfy this equation (such as in a system with another equation), manipulating its form can be crucial. Multiplying both sides of the equation by a constant, including 3, preserves the solution set while often simplifying subsequent calculations.", "---", "### Why Multiply the Equation by 3?", "In many exercises involving systems of equations, one equation may need to match the coefficient format of another for elimination. Multiplying (12x - 3y = 30) by 3 transforms it to:", "[\n3(12x - 3y) = 3 \ imes 30\n]\n[\n36x - 9y = 90\n]", "Now the full system can look like:", "[\n\begin{cases}\n12x - 3y = 30 \\n36x - 9y = 90\n\end{cases}\n]", "---", "### Benefits of This Manipulation", "- Simplifies elimination: The coefficients of (x) and (y) become multiples of each other—here, the second equation’s (x)-coefficient (36) is 3 times that of the first (12), and (y)-coefficient (-9 is 3 times -3). This alignment supports efficient elimination by adding or subtracting equations.\n- Reveals proportional relationships: Multiplying by 3 exposes underlying proportionality between variables, useful in modeling real-world scenarios (e.g., cost per item, rates).\n- Standardizes equation forms: Many algebraic techniques assume cleared denominators or aligned coefficients for streamlined solving.", "---", "### Applying This Transformation in Context", "Suppose you’re solving the system:", "[\n\begin{align}\n12x - 3y &= 30 \quad \ ext{(Equation 1)}\\n12x - 3y &= 90 \quad \ ext{(Equation 2 adjusted via multiplying original by 3)}\n\end{align}\n]", "Subtracting Equation 1 from Equation 2 leads to:", "[\n(36x - 9y) - (12x - 3y) = 90 - 30\n]\n[\n24x - 6y = 60\n]", "This simplified equation reveals a simpler linear relation between (x) and (y), allowing further substitution or elimination.", "---", "### Practical Applications", "This transformation is not purely theoretical:", "- Economics: When comparing cost and revenue models with matching units.\n- Physics: Aligning equations in force and motion calculations.\n- Data Analysis: Preparing data sets for linear regression by standardizing forms.", "---", "### Summary", "Multiplying the equation (12x - 3y = 30) by 3 to obtain (36x - 9y = 90) is a strategic step that prepares the equation for efficient system solving. It aligns coefficients, simplifies elimination techniques, and reveals proportional relationships. Mastering such manipulations strengthens your ability to solve linear systems and supports broader applications across STEM and applied disciplines.", "---", "### Keywords\nmultiply second equation by 3, solve linear equations, 12x - 3y = 30, system of equations, elimination method, linear algebra, algebra steps", "---", "Try it today: Multiply any linear equation by a constant and watch how it streamlines solving simultaneous equations—key to strong problem-solving skills!"]