{"id":10301,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10301"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"word-problems-involving-linear-equations","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/word-problems-involving-linear-equations\/","title":{"rendered":"Word Problems Involving Linear Equations"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Source: Kapdec.com<\/p>\n<\/div>\n

Unit: Linear Equations in One Variable<\/strong><\/h2>\n

Word Problems Involving Linear Equations<\/strong><\/h3>\n

Word problems involving linear equations require translating a real-world situation into a mathematical equation and then solving for the unknown variable. These problems often involve relationships between quantities, rates, and times.<\/p>\n

Steps to Solve Word Problems<\/strong><\/p>\n

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  1. Read the Problem Carefully<\/strong>:\n
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    • Identify what you are asked to find.<\/li>\n
    • Determine the information given.<\/li>\n<\/ul>\n<\/li>\n
    • Define the Variables<\/strong>:\n
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      • Choose a variable to represent the unknown quantity.<\/li>\n
      • Write down what the variable represents.<\/li>\n<\/ul>\n<\/li>\n
      • Set Up the Equation<\/strong>:\n
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        • Translate the words into a mathematical equation using the defined variable.<\/li>\n<\/ul>\n<\/li>\n
        • Solve the Equation<\/strong>:\n
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          • Use algebraic methods to solve for the variable.<\/li>\n<\/ul>\n<\/li>\n
          • Check the Solution<\/strong>:\n
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            • Substitute the solution back into the original context to ensure it makes sense.<\/li>\n
            • Verify the solution meets all conditions of the problem.<\/li>\n<\/ul>\n<\/li>\n
            • Answer the Question<\/strong>:\n
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              • Write a complete sentence answering the problem\u2019s question.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                Examples<\/strong><\/p>\n

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                1. Basic Example<\/strong>:\n
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                  • Problem<\/strong>: A number increased by 7 is 15. What is the number?<\/li>\n
                  • Solution<\/strong>:\n
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                    • Let x<\/em> be the number.<\/li>\n
                    • Equation: x<\/em>+7=15<\/li>\n
                    • Solve: x<\/em>=15\u22127=8<\/li>\n
                    • Answer: The number is 8.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n
                    • Age Problems<\/strong>:\n
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                      • Problem<\/strong>: John is 3 years older than his sister. If the sum of their ages is 23, how old is each?<\/li>\n
                      • Solution<\/strong>:\n
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                        • Let x<\/em> be the sister\u2019s age.<\/li>\n
                        • John\u2019s age: x<\/em>+3<\/li>\n
                        • Equation: x<\/em>+(x<\/em>+3)=23<\/li>\n
                        • Solve: 2\ud835\udc65+3=23<\/li>\n
                        • 2x<\/em>=20<\/li>\n
                        • x<\/em>=10<\/li>\n
                        • Sister\u2019s age: 10<\/li>\n
                        • John\u2019s age: 10+3=13<\/li>\n
                        • Answer: The sister is 10 years old and John is 13 years old.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n
                        • Distance-Rate-Time Problems<\/strong>:\n
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                          • Problem<\/strong>: A car travels at 60 miles per hour. How long will it take to travel 180 miles?<\/li>\n
                          • Solution<\/strong>:\n
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                            • Let t<\/em> be the time in hours.<\/li>\n
                            • Equation: 60t<\/em>=180<\/li>\n
                            • Solve: \ud835\udc61=\n
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                              Source: Kapdec.com<\/p>\n<\/div>\n

                              =3<\/li>\n

                            • Answer: It will take 3 hours to travel 180 miles.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n
                            • Mixture Problems<\/strong>:\n
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                              • Problem<\/strong>: How many liters of a 10% salt solution must be mixed with 20 liters of a 25% salt solution to get a 20% salt solution?<\/li>\n
                              • Solution<\/strong>:\n
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                                • Let x<\/em> be the liters of the 10% solution.<\/li>\n
                                • Salt from 10% solution: 0.10x<\/em><\/li>\n
                                • Salt from 25% solution: 0.25\u00d720=5<\/li>\n
                                • Total salt: 0.20(x<\/em>+20)<\/li>\n
                                • Equation: 0.10x<\/em>+5=0.20(x<\/em>+20)<\/li>\n
                                • Solve: 0.10\ud835\udc65+5=0.20\ud835\udc65+4<\/li>\n
                                • 5\u22124=0.20\ud835\udc65\u22120.10\ud835\udc65<\/li>\n
                                • 1=0.10x<\/em><\/li>\n
                                • x<\/em>=10<\/li>\n
                                • Answer: 10 liters of the 10% salt solution are needed.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n
                                • Investment Problems<\/strong>:\n
                                    \n
                                  • Problem<\/strong>: Maria invests $5000 in two accounts. One account pays 5% interest and the other pays 7%. If the total interest earned is $320, how much is invested in each account?<\/li>\n
                                  • Solution<\/strong>:\n
                                      \n
                                    • Let x<\/em> be the amount invested at 5%.<\/li>\n
                                    • Amount at 7%: 5000\u2212x<\/em><\/li>\n
                                    • Equation: 0.05\ud835\udc65+0.07(5000\u2212\ud835\udc65)=320<\/li>\n
                                    • Solve: 0.05\ud835\udc65+350\u22120.07\ud835\udc65=320<\/li>\n
                                    • \u22120.02\ud835\udc65+350=320<\/li>\n
                                    • \u22120.02\ud835\udc65=\u221230<\/li>\n
                                    • \ud835\udc65=1500<\/li>\n
                                    • Amount at 5%: $1500<\/li>\n
                                    • Amount at 7%: 5000\u22121500=3500<\/li>\n
                                    • Answer: $1500 is invested at 5% and $3500 at 7%.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                                      Summary<\/strong><\/p>\n

                                        \n
                                      • Steps to Solve<\/strong>: Read carefully, define variables, set up the equation, solve, check, and answer.<\/li>\n
                                      • Examples<\/strong>: Include basic number problems, age problems, distance-rate-time problems, mixture problems, and investment problems.<\/li>\n
                                      • Key Skills<\/strong>: Translating words into equations, solving equations, and interpreting solutions in context.<\/li>\n<\/ul>\n

                                        Mastering word problems involves practice and the ability to connect mathematical concepts to real-life scenarios.<\/p>\n

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