{"id":10040,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10040"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"solving-equations-variable-on-both-sides","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/solving-equations-variable-on-both-sides\/","title":{"rendered":"Solving Equations, Variable On Both Sides"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Linear Equation with one Variable<\/strong><\/h2>\n

Chapter: <\/strong>Solving Equations, Variable on both Sides<\/strong><\/h3>\n

Reference: – Understanding Equations with Variables on Both Sides, Applying the Properties of Equality, Combining Like Terms for Simplification, Rearranging and Isolating the Variable, Checking and Verifying Solutions, Real-World Applications of Equations with Variables on Both Sides<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

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  • Understanding Equations with Variables on Both Sides<\/li>\n
  • Combining Like Terms for Simplification<\/li>\n
  • Rearranging and Isolating the Variable<\/li>\n
  • Real-World Applications of Equations with Variables on Both Sides<\/li>\n<\/ul>\n
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    1. Understanding Equations with Variables on Both Sides<\/u><\/strong>\n
        \n
      • Equations can have unknown quantities, called variables, appearing on both sides of the equality sign.<\/li>\n
      • These equations require a systematic approach to isolate the variable and determine its value.<\/li>\n
      • The goal is to express the equation in a simpler form where the variable is only on one side.<\/li>\n<\/ul>\n<\/li>\n
      • Applying the Properties of Equality<\/u><\/strong>\n
          \n
        • The equality of an equation must be maintained throughout the solving process.<\/li>\n
        • The same mathematical operation, whether addition, subtraction, multiplication, or division, must be applied to both sides.<\/li>\n
        • This ensures that the balance of the equation is preserved and that the solution remains valid.<\/li>\n<\/ul>\n<\/li>\n
        • Combining Like Terms for Simplification<\/u><\/strong>\n
            \n
          • Terms that contain the same variable must be grouped together to reduce complexity.<\/li>\n
          • Constant terms (without variables) should also be combined separately to simplify calculations.<\/li>\n
          • This step helps in systematically eliminating unnecessary terms and progressing toward an isolated variable.<\/li>\n<\/ul>\n<\/li>\n
          • Rearranging and Isolating the Variable<\/u><\/strong>\n
              \n
            • Moving variable terms to one side of the equation and constant terms to the other side is necessary.<\/li>\n
            • Using inverse operations allows for the elimination of unwanted terms, helping to isolate the variable.<\/li>\n
            • This step is critical for solving the equation and finding a clear value for the unknown.<\/li>\n<\/ul>\n<\/li>\n
            • Checking and Verifying Solutions<\/u><\/strong>\n
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              • Once a solution is obtained, it must be substituted back into the original equation to confirm its correctness.<\/li>\n
              • If both sides of the equation result in the same value, the solution is valid.<\/li>\n
              • This step prevents errors and ensures that the solution satisfies the given equation.<\/li>\n<\/ul>\n<\/li>\n
              • Real-World Applications of Equations with Variables on Both Sides<\/u><\/strong>\n
                  \n
                • Such equations frequently appear in real-life situations, such as budgeting, distance-speed-time problems, and scientific calculations.<\/li>\n
                • Understanding their applications enhances problem-solving skills and logical reasoning.<\/li>\n
                • Mastering these equations helps in fields like finance, engineering, and physics, where precise calculations are essential.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                  Example: –<\/strong><\/p>\n

                  A company offers two different mobile data plans:<\/p>\n

                    \n
                  • Plan A charges a fixed monthly fee of $20 plus $2 per GB of data used.<\/li>\n
                  • Plan B has no fixed fee but charges $4 per GB of data used.<\/li>\n<\/ul>\n

                    A customer wants to determine after how many GB of data usage the cost of both plans will be the same.<\/p>\n

                    Using equations with variables on both sides, simplification techniques, and verification, find:<\/p>\n

                      \n
                    1. The number of GB at which both plans cost the same.<\/li>\n
                    2. Verify the solution by substituting it back into the equation.<\/li>\n<\/ol>\n

                      Solution: –<\/u><\/strong>
                      \n <\/p>\n

                      Step 1: Forming the Equation<\/u><\/p>\n

                      Let x be the number of GB of data used.<\/p>\n

                        \n
                      • Cost of Plan A = Fixed fee + (Cost per GB × Data usage)<\/li>\n<\/ul>\n

                        20+2x<\/p>\n

                          \n
                        • Cost of Plan B = Cost per GB × Data usage<\/li>\n<\/ul>\n

                          4x<\/p>\n

                          Since we are looking for the point where both plans cost the same:<\/p>\n

                          20+2x=4x<\/p>\n

                           <\/p>\n

                          Step 2: Solving the Equation<\/u><\/p>\n

                          Rearrange and isolate the variable by moving all terms with x to one side:<\/p>\n

                          20=4x−2x<\/p>\n

                          Divide both sides by 2:<\/p>\n

                          x=20\/2 = 10<\/p>\n

                          Thus, the data usage at which both plans cost the same is 10 GB.<\/p>\n

                           <\/p>\n

                          Step 3: Verifying the Solution<\/u><\/p>\n

                          Substituting x = 10 back into both cost equations:<\/p>\n

                            \n
                          • Plan A cost<\/li>\n<\/ul>\n

                            20+2(10) =20+20=40<\/p>\n

                              \n
                            • Plan B cost<\/li>\n<\/ul>\n

                              4(10) =40<\/p>\n

                              Since both costs are equal ($40), our solution is verified.<\/p>\n

                              Conclusive Points for "Solving Equations with Variables on Both Sides"<\/u><\/strong><\/p>\n

                                \n
                              1. Equations with variables on both sides require systematic simplification<\/strong>\n
                                  \n
                                • To solve such equations, the variable terms must be brought to one side, while constant terms are moved to the other.<\/li>\n<\/ul>\n<\/li>\n
                                • Maintaining equality is crucial in every step<\/strong>\n
                                    \n
                                  • Applying the same mathematical operations to both sides ensure the balance of the equation remains unchanged.<\/li>\n<\/ul>\n<\/li>\n
                                  • Combining like terms simplifies the equation<\/strong>\n
                                      \n
                                    • Grouping similar terms helps in reducing complexity and making the equation easier to solve.<\/li>\n<\/ul>\n<\/li>\n
                                    • Checking the solution ensures accuracy<\/strong>\n
                                        \n
                                      • Substituting the obtained value back into the original equation helps verify correctness and avoid mistakes.<\/li>\n<\/ul>\n<\/li>\n
                                      • Such equations have practical applications in real-world problem-solving<\/strong>\n
                                          \n
                                        • These equations are used in business, physics, finance, and engineering, making them an essential mathematical concept.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

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