{"id":10302,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10302"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"solving-basic-linear-equations","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/solving-basic-linear-equations\/","title":{"rendered":"Solving Basic 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

Solving Basic Linear Equations<\/strong><\/h3>\n

Linear equations in one variable are fundamental in algebra and represent the simplest form of equations. They are used to model and solve real-world problems involving relationships between quantities.<\/p>\n

Definition and Standard Form<\/strong><\/p>\n

A linear equation in one variable can be written in the standard form:<\/p>\n

ax <\/em>+ b<\/em>=0 where a<\/em> and b<\/em> are constants, and x<\/em> is the variable.<\/p>\n

Solving Linear Equations<\/strong><\/p>\n

To solve a linear equation, the goal is to isolate the variable on one side of the equation. Here are the steps:<\/p>\n

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  1. Simplify both sides of the equation:<\/strong>\n
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    • Combine like terms.<\/li>\n
    • Remove parentheses by using the distributive property.<\/li>\n<\/ul>\n<\/li>\n
    • Move the variable term to one side:<\/strong>\n
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      • Use addition or subtraction to get all variable terms on one side and constant terms on the other side.<\/li>\n<\/ul>\n<\/li>\n
      • Isolate the variable:<\/strong>\n
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        • Divide or multiply to solve for the variable.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

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

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          1. Basic Example:<\/strong> 3x<\/em>+5=11\n
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            • Subtract 5 from both sides: 3x<\/em>=6<\/li>\n
            • Divide both sides by 3: x<\/em>=2<\/li>\n<\/ul>\n<\/li>\n
            • With Fractions:<\/strong>\n
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              Source: Kapdec.com<\/p>\n<\/div>\n

              \u22124=8<\/p>\n

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              • Add 4 to both sides:\n
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                =12<\/li>\n

              • Multiply both sides by 3: 2x<\/em>=36<\/li>\n
              • Divide both sides by 2: x<\/em>=18<\/li>\n<\/ul>\n<\/li>\n
              • Variable on Both Sides:<\/strong> 4\ud835\udc65\u22127=2\ud835\udc65+5\n
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                • Subtract 2x<\/em> from both sides: 2x<\/em>\u22127=5<\/li>\n
                • Add 7 to both sides: 2x<\/em>=12<\/li>\n
                • Divide both sides by 2: x<\/em>=6<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                  Special Cases<\/strong><\/p>\n

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                  1. No Solution:<\/strong> If simplifying the equation leads to a contradiction (e.g., 0=5), the equation has no solution.<\/li>\n<\/ol>\n

                    2x<\/em>+3=2x<\/em>+7<\/p>\n

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                    1. \n
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                      • Subtract 2x<\/em> from both sides: 3=7<\/li>\n
                      • This is a contradiction, so there is no solution.<\/li>\n<\/ul>\n<\/li>\n
                      • Infinite Solutions:<\/strong> If simplifying the equation results in a tautology (e.g., 0=0), the equation has infinitely many solutions. 3(x<\/em>\u22121)=3x<\/em>\u22123\n
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                        • Distribute and simplify: 3x<\/em>\u22123=3x<\/em>\u22123<\/li>\n
                        • Subtract 3\ud835\udc653x<\/em> from both sides: \u22123=\u22123<\/li>\n
                        • This is always true, so there are infinitely many solutions.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                          Applications<\/strong><\/p>\n

                          Linear equations in one variable are used in various real-life applications such as:<\/p>\n

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                          1. Solving for unknown quantities:<\/strong>\n
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                            • Example: If the total cost C<\/em> is given by C<\/em>=5x<\/em>+20, where x<\/em> is the number of items, and you know the total cost, you can solve for x<\/em>.<\/li>\n<\/ul>\n<\/li>\n
                            • Solving for rates:<\/strong>\n
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                              • Example: If distance d<\/em> travelled is given by d<\/em>=rt<\/em> (rate r<\/em> times time t<\/em>), and you know d<\/em> and r<\/em>, you can solve for t<\/em>.<\/li>\n<\/ul>\n<\/li>\n
                              • Budgeting and finance:<\/strong>\n
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                                • Example: If your total monthly expenses E<\/em> are given by E<\/em>=200x<\/em>+500, where x<\/em> is the number of utility units used, and you know your budget, you can solve for x<\/em>.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

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

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                                  • Definition:<\/strong> Linear equations in one variable are of the form ax<\/em>+b<\/em>=0.<\/li>\n
                                  • Solving Steps:<\/strong> Simplify both sides, move variable terms to one side, isolate the variable.<\/li>\n
                                  • Special Cases:<\/strong> Recognize no solution (contradiction) and infinite solutions (tautology).<\/li>\n
                                  • Applications:<\/strong> Used in various real-life contexts such as solving for unknown quantities, rates, and financial calculations.<\/li>\n<\/ul>\n

                                    Mastering linear equations in one variable is essential for building a strong foundation in algebra and solving more complex mathematical problems.<\/p>\n

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