Solve the system of equations: \( 3x + 4y = 10 \) and \( 2x - y = 3 \).

["Solving a System of Equations: How to Solve ( 3x + 4y = 10 ) and ( 2x - y = 3 )", "Solving a system of equations is a fundamental skill in algebra, essential for fields ranging from engineering to economics. Today, we’ll walk through a clear and step-by-step method to solve the system:\n[\n\begin{cases}\n3x + 4y = 10 \quad \ ext{(Equation 1)} \\n2x - y = 3 \quad \ ext{(Equation 2)}\n\end{cases}\n]", "Understanding how to solve systems of equations opens doors to finding real-world solutions, such as determining optimal resource allocation or analyzing intersecting trends.", "---", "### Step 1: Choose a Method to Solve the System", "There are several approaches—substitution, elimination, and graphing. Here, we’ll focus on the substitution method, which works well for this pair of linear equations.", "From Equation 2, solve for one variable in terms of the other:", "[\n2x - y = 3 \Rightarrow y = 2x - 3\n]", "---", "### Step 2: Substitute into the First Equation", "Now, substitute ( y = 2x - 3 ) into Equation 1:", "[\n3x + 4y = 10\n\Rightarrow 3x + 4(2x - 3) = 10\n]", "Simplify:", "[\n3x + 8x - 12 = 10\n\Rightarrow 11x - 12 = 10\n]", "Add 12 to both sides:", "[\n11x = 22\n]", "Divide by 11:", "[\nx = 2\n]", "---", "### Step 3: Solve for the Other Variable", "Now substitute ( x = 2 ) back into the expression for ( y ):", "[\ny = 2x - 3 = 2(2) - 3 = 4 - 3 = 1\n]", "---", "### Step 4: Verify the Solution", "Plug ( x = 2 ) and ( y = 1 ) into both original equations to confirm correctness:", "Equation 1:\n( 3(2) + 4(1) = 6 + 4 = 10 ) ✅", "Equation 2:\n( 2(2) - 1 = 4 - 1 = 3 ) ✅", "Both equations are satisfied, confirming our solution is correct.", "---", "### The Solution", "The solution to the system is:\n[\n\boxed{(x, y) = (2, 1)}\n]", "---", "### Why This Matters", "Solving systems of equations helps model real-life problems, such as finding the break-even point in economics or modeling intersecting motion paths in physics. Mastering techniques like substitution builds a strong foundation in algebra and beyond.", "If you’re studying math, understanding structured problem-solving is key—this method applies to any pair of linear equations.", "---", "Key Takeaways:\n- Use substitution when one equation is easily solved for a variable.\n- Always verify your solution by plugging values back into original equations.\n- The solution ((2, 1)) is a point where both lines intersect.", "Start practicing systems like this daily—soon, solving equations will feel intuitive!"]