Writing and Solving Linear Equations from Math Sentences - GED Math

Translating Math Sentences into Equations

• 🎯 The problem involves translating a complex math sentence into a linear equation, using keywords to guide the setup.
• 🔑 Keywords like "quotient" indicate division (a fraction), "decreased by" indicates subtraction, "difference" indicates subtraction, and "is" indicates equality.
• 💡 The sentence "The quotient of seven times a number decreased by 19 and the difference of twice a number and five is three" is broken down into its components.

Setting Up the Linear Equation

• ✍️ "Seven times a number decreased by 19" is represented as 7x - 19.
• ➕ "The difference of twice a number and five" is represented as 2x - 5.
• ➗ The "quotient" of these two expressions means (7x - 19) / (2x - 5).
• ⚖️ The phrase "is three" translates to = 3, forming the equation (7x - 19) / (2x - 5) = 3.

Solving the Linear Equation

• ✖️ To solve, cross-multiplication is used: 3 * (2x - 5) = 1 * (7x - 19).
• 🧮 Distributing the 3 gives 6x - 15 = 7x - 19.
• ➡️ To isolate x, subtract 6x from both sides, resulting in -15 = x - 19.
• ➕ Adding 19 to both sides yields 4 = x.
• ✅ The number is 4, which is the solution to the problem.