Equations: \( 5n + 3p = 23 \) and \( 3n + 2p = 14 \)
["Understanding the Equations: ( 5n + 3p = 23 ) and ( 3n + 2p = 14 )", "Solving systems of linear equations is a fundamental skill in algebra, connecting mathematics to real-world applications such as economics, physics, and engineering. Two common types of problems involve equations like ( 5n + 3p = 23 ) and ( 3n + 2p = 14 ), where ( n ) and ( p ) represent unknown variables. This article explores how to solve these equations using substitution and elimination methods, explain their significance, and highlight practical applications.", "### The System of Equations", "We are given:", "[\n\begin{cases}\n5n + 3p = 23 \quad \ ext{(1)} \\n3n + 2p = 14 \quad \ ext{(2)}\n\end{cases}\n]", "These equations can represent relationships between two variables, such as the number of items ( n ) and ( p ) in a cost model or physical quantities in a system.", "### Step 1: Use the Elimination Method", "To eliminate one variable and solve for the other, choose a variable to eliminate. Here, we eliminate ( p ):", "- Multiply equation (1) by 2:\n ( 2(5n + 3p) = 2 \ imes 23 \Rightarrow 10n + 6p = 46 ) — (3)", "- Multiply equation (2) by 3:\n ( 3(3n + 2p) = 3 \ imes 14 \Rightarrow 9n + 6p = 42 ) — (4)", "Now subtract equation (4) from equation (3):", "[\n(10n + 6p) - (9n + 6p) = 46 - 42\n]", "[\n10n + 6p - 9n - 6p = 4 \Rightarrow n = 4\n]", "### Step 2: Solve for the Other Variable", "Substitute ( n = 4 ) into one of the original equations. Using equation (2):", "[\n3(4) + 2p = 14 \Rightarrow 12 + 2p = 14\n]", "[\n2p = 14 - 12 = 2 \Rightarrow p = 1\n]", "### Solution", "The solution to the system is:", "[\nn = 4, \quad p = 1\n]", "### Why This Equation System Matters", "Solving systems like ( 5n + 3p = 23 ) and ( 3n + 2p = 14 ) helps model real-life situations:", "- Business Applications: Determining pricing and quantities in profit maximization.\n- Science: Balancing chemical equations or electrical circuits.\n- Finance: Analyzing loan interest or investment growth.", "These equations form a linear model where each line represents a constraint, and their intersection defines a unique outcome.", "### Key Takeaways", "- Substitution and elimination are powerful techniques for solving simultaneous equations.\n- Always verify by substituting back into original equations to ensure accuracy.\n- Systems of equations model real-world scenarios, making abstract math applicable.", "### Next Steps", "- Practice solving with different coefficients to build fluency.\n- Explore graphical methods to visualize solutions on a coordinate plane.\n- Apply these concepts to coding problems or real-life decision-making models.", "---", "Understanding how to solve ( 5n + 3p = 23 ) and ( 3n + 2p = 14 ) opens the door to mastering more complex algebraic systems and their applications across disciplines."]
