To solve, subtract the first equation from the second:
["Title: Master Algebra: Solving Simultaneous Equations by Subtraction – A Step-by-Step Guide", "When learning algebra, one of the most powerful techniques for solving simultaneous equations is subtraction or elimination—essentially, subtracting one equation from another to eliminate a variable. This method simplifies systems of equations and leads to clear, concise solutions. In this article, we’ll explore how to solve simultaneous equations by subtracting the first from the second, explain why this works, and provide step-by-step examples to help learners master this essential algebraic skill.", "---", "### What Are Simultaneous Equations?", "Simultaneous equations are two or more equations with the same variables that are solved together. For example:", "[\n\begin{cases}\nx + y = 10 \quad \ ext{(Equation 1)} \\n2x - y = 5 \quad \ ext{(Equation 2)}\n\end{cases}\n]", "These equations represent relationships between variables (x) and (y), and our goal is to find their exact values.", "---", "### Why Subtract Equations to Solve?", "Subtracting one equation from another eliminates one variable, reducing the system to a single equation with one unknown—much easier to solve. This technique leverages the principle of consistency in systems: if two equations describe the same point in the coordinate plane, subtracting them eliminates redundancy.", "---", "### Step-by-Step: Subtracting the First from the Second Equation", "Let’s break down the process clearly with an example:", "Step 1: Write down both equations clearly", "[\n\begin{align}\n(1)\quad & x + y = 10 \\n(2)\quad & 2x - y = 5\n\end{align}\n]", "Step 2: Choose the variable to eliminate", "Here, we aim to eliminate (y). Notice that in Equation (1), (y) has a coefficient of +1, and in Equation (2), (y) has a coefficient of –1. Subtracting Equation (1) from Equation (2) cancels (y):", "[\n(2x - y) - (x + y) = 5 - 10\n]", "Step 3: Perform the subtraction", "[\n2x - y - x - y = 5 - 10\n]\n[\n(2x - x) + (-y - y) = -5\n]\n[\nx - 2y = -5\n]", "Wait — actually, in standard elimination, we subtract left sides from left sides and right sides from right sides, so let’s double-check properly:", "[\n(2x - y) - (x + y) = 5 - 10\n]\n[\n2x - y - x - y = -5\n]\n[\n(2x - x) + (-y - y) = -5\n]\n[\nx - 2y = -5 \quad \ ext{(This is a new equation)}\n]", "But this introduces a new equation with both (x) and (y), which is not helpful for elimination—because we now lose the direct (x = \dots) form.", "Alternative (better) approach: Add instead? No — subtraction still works if we align variables properly.", "Let’s return to the goal: eliminate (y) by aligning coefficients.", "Actually, to eliminate (y), notice:", "- Equation (1) has (+y)\n- Equation (2) has (-y), so subtracting Equation (1) from Equation (2):", "[\n(2x - y) - (x + y) = 5 - 10\n]\n[\n2x - y - x - y = -5\n]\n[\nx - 2y = -5 \quad \ ext{(still has } y\ ext{)}\n]", "So subtraction alone doesn’t eliminate (y) here — unless the coefficients on (y) are opposite. Let’s adjust our example so elimination works cleanly.", "---", "### Improved Example with Clear Elimination", "Let’s consider:", "[\n\begin{cases}\n2x + 3y = 12 \quad \ ext{(Eq 1)}\\nx - 3y = 3 \quad \ ext{(Eq 2)}\n\end{cases}\n]", "Now, observe: the (y)-terms are (+3y) and (-3y)—opposite signs. This is perfect for elimination by addition or subtraction.", "To eliminate (y), add the equations:", "[\n(2x + 3y) + (x - 3y) = 12 + 3\n]\n[\n2x + x + 3y - 3y = 15\n]\n[\n3x = 15 \quad \Rightarrow \quad x = 5\n]", "Now substitute (x = 5) into one equation to find (y):", "[\n5 - 3y = 3 \Rightarrow -3y = -2 \Rightarrow y = \frac{2}{3}\n]", "But the question was: Solve by subtracting the first equation from the second. So we add in this case — but subtraction is still a valid elimination technique depending on setup.", "However, subtracting works best when subtracting variables cancel.", "Let’s redo with a clean case where subtraction eliminates one variable cleanly.", "---", "### Clean Example Designed for Subtraction", "Consider:", "[\n\begin{cases}\n3x + 2y = 16 \quad \ ext{(Eq 1)}\\n3x - 2y = 4 \quad \ ext{(Eq 2)}\n\end{cases}\n]", "Here, (3x) has coefficient 3 in both. Subtract Equation (2) from Equation (1):", "[\n(3x + 2y) - (3x - 2y) = 16 - 4\n]\n[\n3x + 2y - 3x + 2y = 12\n]\n[\n4y = 12 \Rightarrow y = 3\n]", "Now substitute back:", "[\n3x + 2(3) = 16 \Rightarrow 3x + 6 = 16 \Rightarrow 3x = 10 \Rightarrow x = \frac{10}{3}\n]", "✅ Clear solution: (x = \frac{10}{3}, y = 3)", "Key takeaway: By subtracting Eq (2) from Eq (1), the (3x) terms cancel, isolates (2y), and enables easy solving.", "---", "### When Does Subtraction Eliminate a Variable?", "Subtracting equations eliminates a variable when:", "- The coefficients on that variable are opposite (e.g., (+y) and (-y))\n- Or when rearranging equations flips signs so coefficients cancel.", "Always check variable signs before subtracting.", "---", "### Step-by-Step Summary", "1. Write both equations clearly, ensuring variables are aligned.\n2. Arrange equations so like terms line up.\n3. Subtract one equation from the other, removing a variable due to opposing coefficients.\n4. Solve the resulting equation for the remaining variable.\n5. Substitute back to find the other variable.\n6. Verify by plugging values into both original equations.", "---", "### Why This Method Matters", "- Efficiency: Directly reduces system complexity.\n- Clarity: Minimizes risks of sign errors compared to substitution in complex systems.\n- Scalability: Works for systems with three or more variables when applied systematically.", "---", "### Real-World Applications", "Solving simultaneous equations by elimination/subtraction appears in:", "- Economics (supply and demand models)\n- Physics (optimization problems)\n- Engineering (system load balances)\n- Business (break-even analysis)", "Understanding this method empowers students and professionals to solve real-world problems analytically.", "---", "### Final Tips for Success", "- Represent equations clearly (order of terms matters).\n- Write expansions of subtracted expressions carefully.\n- Always simplify after subtraction.\n- Practice with varied coefficient signs to master cancellation.", "---", "Conclusion\nSubtracting equations to solve simultaneous systems is a foundational algebraic technique that strengthens problem-solving skills. By carefully aligning terms and leveraging opposite coefficients, learners can eliminate variables efficiently and solve complex systems with confidence. Master this method—it’s your key to unlocking advanced math and real-world applications.", "---", "Frequently Asked Questions (FAQs)", "Q: Why not always use substitution instead of elimination?\nA: Substitution is ideal when one variable is easily solved for; elimination is generally faster for systems with clear coefficient matches.", "Q: Can I subtract equations in any order?\nA: Yes, but choose the order that produces cancellation of the variable you want to eliminate—usually the one with larger coefficient or opposite signs.", "Q: What if subtraction doesn’t eliminate a variable?\nA: Rearrange one or both equations (e.g., multiply by constants) so opposing terms line up cleanly.", "---", "Start practicing today—solve one system by subtraction, and watch your algebra skills soar!"]
