Multiply the second equation by 2 to eliminate \( y \):

["Multiply the Second Equation by 2 to Eliminate ( y ): A Step-by-Step Guide for Solving Systems of Equations", "When solving systems of linear equations, eliminating one variable can simplify the process significantly. One common technique is to multiply one equation by a scalar to align coefficients, allowing substitution and elimination. Here’s how multiplying the second equation by 2 can effectively eliminate ( y ) and streamline solving:", "### Why Eliminate ( y )?\nIn many systems of equations, eliminating a variable like ( y ) reduces the problem to a single-variable equation, making it easier to solve. Once ( y ) is eliminated, you substitute its value back into one of the original equations to find ( x ).", "### Example Problem\nConsider the system:\n[\n\begin{cases}\nx + 2y = 10 \quad \ ext{(1)} \\n2x + y = 7 \quad \ ext{(2)}\n\end{cases}\n]", "Notice that if we multiply equation (1) by 2, the ( y )-terms become ( +4y ) and ( +4y ), which can later be subtracted to eliminate ( y ).", "### Step 1: Multiply the second equation by 2\nStart by targeting the variable to eliminate. Since ( y ) appears with coefficient 2 in equation (1) and 1 in equation (2), multiplying equation (2) by 2 gives:\n[\n2 \ imes (2x + y) = 2 \ imes 7 \Rightarrow 4x + 2y = 14 \quad \ ext{(2')}\n]", "Now, the system becomes:\n[\n\begin{aligned}\nx + 2y &= 10 \quad \ ext{(1)} \\n4x + 2y &= 14 \quad \ ext{(2')}\n\end{aligned}\n]", "### Step 2: Subtract Equation (1) from (2') to eliminate ( y )\nSubtract equation (1) from equation (2'):\n[\n(4x + 2y) - (x + 2y) = 14 - 10\n]\n[\n4x + 2y - x - 2y = 4\n]\n[\n3x = 4\n]\n[\nx = \frac{4}{3}\n]", "### Step 3: Substitute ( x = \frac{4}{3} ) into one original equation\nBack-substitute into equation (1):\n[\n\frac{4}{3} + 2y = 10\n]\n[\n2y = 10 - \frac{4}{3} = \frac{30}{3} - \frac{4}{3} = \frac{26}{3}\n]\n[\ny = \frac{13}{3}\n]", "### Final Solution\nThe solution to the system is ( x = \frac{4}{3} ), ( y = \frac{13}{3} ). By multiplying the second equation by 2 first, we aligned ( y )-coefficients for efficient elimination—turning a paired system into two solvable one-variable equations.", "### Bonus Tip: When to Use Conversion Instead\nSometimes, multiplying hands-on is quicker; other times, swapping equations first makes multiplication strategic. Understanding both approaches strengthens algebraic flexibility.", "---", "Key Takeaways:\n- Multiply one equation by a scalar to match coefficients of the target variable.\n- Aligning terms allows direct elimination when adding/subtracting equations.\n- This method simplifies solving systems into manageable steps.", "This technique is especially useful in graphing, substitution, and elimination methods. Practice with varied systems to master eliminating variables effortlessly! \nKeywords: eliminate ( y ), multiply equation by 2, elimination method, solve linear systems, eliminate variable, algebraic technique, step-by-step elimination."]