<h3>The Main Idea</h3> <p><strong>Two unknowns, two relationships — turn the words into two equations, then solve.</strong></p> <p>Systems of equations problems give you information about two things (coffee and tea, watches and bracelets). Your job is to translate that information into math, then use substitution or elimination to find the answer.</p> <h3>How to Recognize a System</h3> <p>You're dealing with a system when the problem gives you <strong>two relationships involving the same two unknowns</strong>:</p> <ul> <li>Relationship 1: A combination with a total (e.g., "3 coffees and 6 teas cost $33")</li> <li>Relationship 2: How the items relate to each other (e.g., "tea costs $2 less than coffee")</li> </ul> <p>This is NOT proportions — you're not comparing the same things in the same way. You're combining them in different ways.</p> <h3>Two Ways to Solve</h3> <p><strong>Substitution:</strong> When one equation is already solved for a variable (like t = c - 2), plug that expression into the other equation.</p> <p><strong>Elimination:</strong> When both equations have the same structure (like a + b = 44 and a - b = 8), add or subtract the equations to cancel out one variable.</p> <ul> <li>Use <strong>substitution</strong> when you see "y = ..." or "x = ..." already isolated</li> <li>Use <strong>elimination</strong> when you see matching structures like (a + b) and (a - b)</li> </ul> <h3>The Approach</h3> <ol> <li><strong>Read the question first</strong> — What do you want? (tea or coffee? older or younger?)</li> <li><strong>Define variables</strong> — Use letters that make sense (c for coffee, t for tea)</li> <li><strong>Write equation 1</strong> — The combination or sum</li> <li><strong>Write equation 2</strong> — The relationship or difference</li> <li><strong>Choose your method</strong> — Substitution or elimination</li> <li><strong>Solve</strong> — Get one variable, then find the other</li> <li><strong>Answer what was asked</strong> — If you solved for c but they want t, do one more step!</li> </ol> <h3>Common Pitfall</h3> <p><strong>Solving for the wrong variable.</strong> You might solve for coffee when they asked for tea. Always check the question sentence before selecting your answer.</p>
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