Solve the system of equations: \(2x + 3y = 12\) and \(x - y = 3\).
["# Solve the System of Equations: (2x + 3y = 12) and (x - y = 3)", "Solving systems of linear equations is a fundamental skill in algebra, widely used in mathematics, science, and real-life problem-solving. In this article, we’ll walk through how to solve the system of equations:", "[\n\begin{cases}\n2x + 3y = 12 \\nx - y = 3\n\end{cases}\n]", "## Why Solve Systems of Equations?", "Understanding how to solve simultaneous equations helps model real-world scenarios—such as budget planning, physics problems, and business modeling—by finding consistent solutions for multiple constraints.", "## Step-by-Step Solution", "We’ll solve the system using the substitution method, one of the most efficient techniques for solving such equations.", "### Step 1: Solve one equation for one variable", "Start with the second equation because it’s simpler:", "[\nx - y = 3\n]", "Solve for (x):", "[\nx = y + 3\n]", "### Step 2: Substitute into the second equation", "Now substitute (x = y + 3) into the first equation (2x + 3y = 12):", "[\n2(y + 3) + 3y = 12\n]", "### Step 3: Simplify and solve for (y)", "Expand and combine like terms:", "[\n2y + 6 + 3y = 12 \\n5y + 6 = 12\n]", "Subtract 6 from both sides:", "[\n5y = 6\n]", "Divide by 5:", "[\ny = \frac{6}{5}\n]", "### Step 4: Substitute back to find (x)", "Recall from earlier: (x = y + 3). Substitute (y = \frac{6}{5}):", "[\nx = \frac{6}{5} + 3 = \frac{6}{5} + \frac{15}{5} = \frac{21}{5}\n]", "### Step 5: Write the solution", "The solution to the system is:", "[\nx = \frac{21}{5}, \quad y = \frac{6}{5}\n]", "### Verification", "Plug the values into both original equations to confirm:", "1. (2x + 3y = 2\left(\frac{21}{5}\right) + 3\left(\frac{6}{5}\right) = \frac{42}{5} + \frac{18}{5} = \frac{60}{5} = 12) ✔️\n2. (x - y = \frac{21}{5} - \frac{6}{5} = \frac{15}{5} = 3) ✔️", "Both equations are satisfied.", "## Conclusion", "Solving (2x + 3y = 12) and (x - y = 3) yields the unique solution:", "[\n\boxed{x = \frac{21}{5},\ y = \frac{6}{5}}\n]", "Mastering this method opens the door to solving more complex systems and applies broadly in science, engineering, and economics.", "---", "Keywords: solve system of equations, solve (2x + 3y = 12) and (x - y = 3), substitution method, linear algebra, step-by-step solution, algebra tutorial, mathematical problem solving."]
