Solve the system of equations: \( 3x + 2y = 12 \) and \( 5x - y = 7 \).

["# Solve the System of Equations: ( 3x + 2y = 12 ) and ( 5x - y = 7 )", "Solving systems of equations is a foundational skill in algebra, widely used in science, engineering, economics, and everyday problem-solving. Whether you're predicting business models or optimizing resources, mastering how to find the values of ( x ) and ( y ) that satisfy multiple conditions is essential. One common and useful example is solving the equations:", "[\n\begin{cases}\n3x + 2y = 12 \\n5x - y = 7\n\end{cases}\n]", "This article explains how to solve this system step-by-step, using two effective methods: substitution and elimination, helping you build confidence in finding exact solutions for linear equations.", "---", "## Why Solve Systems of Equations?", "Before diving into the solution, understanding why we solve systems is valuable. Real-world problems often involve multiple constraints—like budget limits, resource capacities, or trade-offs between variables. Systems of equations provide precise answers by balancing these constraints mathematically, ensuring solutions align with all conditions simultaneously.", "---", "## Method 1: Substitution Method", "The substitution method involves solving one equation for a variable, then substituting that expression into the other equation. This works well when one variable is easily isolated.", "Step 1: Solve one equation for ( y )\nStart with the second equation, which has the simplest coefficient for ( y ):\n[\n5x - y = 7\n]\nAdd ( y ) and subtract 7 from both sides:\n[\n-y + 5x = 7 \implies y = 5x - 7\n]", "Step 2: Substitute into the first equation\nNow replace ( y ) in the first equation ( 3x + 2y = 12 ) using ( y = 5x - 7 ):\n[\n3x + 2(5x - 7) = 12\n]\nDistribute the 2:\n[\n3x + 10x - 14 = 12\n]\nCombine like terms:\n[\n13x - 14 = 12\n]\nAdd 14 to both sides:\n[\n13x = 26\n]\nDivide by 13:\n[\nx = 2\n]", "Step 3: Solve for ( y )\nUse ( x = 2 ) in the expression ( y = 5x - 7 ):\n[\ny = 5(2) - 7 = 10 - 7 = 3\n]", "Solution: ( (x, y) = (2, 3) )", "---", "## Method 2: Elimination Method", "When equations have convenient coefficients for addition or subtraction, elimination is efficient. Here, we’ll eliminate ( y ) by manipulating the equations properly.", "Step 1: Align equations for elimination\nThe system is:\n[\n\begin{aligned}\n(1)&\quad 3x + 2y = 12 \\n(2)&\quad 5x - y = 7\n\end{aligned}\n]\nTo eliminate ( y ), make the coefficients of ( y ) opposites. Multiply equation (2) by 2:\n[\n2(5x - y) = 2(7) \implies 10x - 2y = 14\n]\nNow rewrite the system:\n[\n\begin{aligned}\n3x + 2y &= 12 \\n10x - 2y &= 14\n\end{aligned}\n]", "Step 2: Add the equations\nAdding both equations cancels ( y ):\n[\n(3x + 10x) + (2y - 2y) = 12 + 14 \implies 13x = 26\n]\n[\nx = 2\n]", "Step 3: Back-substitute to find ( y )\nUse ( x = 2 ) in equation (2):\n[\n5(2) - y = 7 \implies 10 - y = 7\n]\n[\ny = 10 - 7 = 3\n]", "Result: ( (x, y) = (2, 3) )", "---", "## Verifying the Solution", "Always verify your solution by plugging ( x = 2 ) and ( y = 3 ) into both original equations:", "- First equation: ( 3(2) + 2(3) = 6 + 6 = 12 ) ✓\n- Second equation: ( 5(2) - 3 = 10 - 3 = 7 ) ✓", "Since both equations are satisfied, the solution is correct.", "---", "## Real-World Applications", "This type of problem-solving applies to budgets, chemical mixtures, motion problems, and more. For example, suppose ( x ) and ( y ) represent quantities of two ingredients with cost and volume constraints. Solving the system determines exactly how much to use to meet both conditions.", "---", "## Key Takeaways", "- Solving systems finds values satisfying multiple equations simultaneously.\n- Substitution works well when a variable is easily isolated.\n- Elimination eliminates a variable through strategic addition/subtraction.\n- Verifying by plugging values confirms correctness.", "Mastering these methods builds confidence in tackling linear systems, a critical tool across science, technology, and daily decision-making.", "---", "Try it now! With practice, solving equations by substitution or elimination becomes intuitive—great for exams, homework, or real-life challenges.", "---", "Keywords: solve system of equations, linear equations, substitution method, elimination method, algebra practice, step-by-step solution, real-world applications, x and y system"]