{"id":9876,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9876"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"solving-equations-variable-on-both-side","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/solving-equations-variable-on-both-side\/","title":{"rendered":"Solving Equations, Variable On Both Side"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Algebra – 1<\/strong><\/h2>\n

Chapter: <\/strong>Solving Equations, Variable on Both Side<\/strong><\/h3>\n

Reference: – Introduction to Variable on Both Sides, Collecting Like Terms, Moving Variables to One Side, Balancing Method, Transposition Method, Equations with Parentheses, Equations with Fractions and Decimals, Special Cases (No Solution, Infinite Solutions), Verification, Solved Examples, Odd-One-Out Problems, Common Mistakes, Practice Grid<\/em><\/p>\n

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

    \n
  • Introduction to Linear Equation in One Variable<\/em><\/li>\n
  • Solving Equation on Variable on Both Side<\/em><\/li>\n
  • Collecting Variable Terms on Both side<\/em><\/li>\n
  • Equation with Parentheses, Fractions & Decimals <\/em><\/li>\n<\/ul>\n

    Introduction to Variable on Both Sides<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    In many linear equations, the variable appears on\u00a0both sides\u00a0of the equal\u2019s sign. To solve such equations, we must first collect all variable terms on one side and all constant terms on the other.<\/p>\n

    General Form:<\/strong>\u00a0ax + b = cx + d (where a, b, c, d are constants, a \u2260 c or possibly a = c)<\/p>\n

    When we solve equations with variable on both sides, we essentially ask:<\/p>\n

    “How do I get all the x’s together on one side so I can isolate the variable?”<\/p>\n

    The answer:\u00a0Add or subtract variable terms from both sides\u00a0to move them.<\/p>\n

    Importance<\/u><\/strong><\/p>\n

      \n
    • Builds on basic equation-solving skills<\/li>\n
    • Essential for solving real-world problems where variables appear in multiple places<\/li>\n
    • Foundation for systems of equations and advanced algebra<\/li>\n<\/ul>\n

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

      Equation:<\/strong>\u00a03x + 2 = x + 10
      \nStep 1:<\/strong>\u00a0Subtract x from both sides \u2192 2x + 2 = 10
      \nStep 2:<\/strong>\u00a0Subtract 2 from both sides \u2192 2x = 8
      \nStep 3:<\/strong>\u00a0Divide by 2 \u2192 x = 4<\/p>\n

      So, if we had 3x + 2 = 3x + 5, that simplifies to 2 = 5 (no solution).<\/p>\n

      Subtopics<\/u><\/strong><\/p>\n

      1. Why Move Variables to One Side?<\/strong><\/p>\n

      Having variables on both sides makes it impossible to isolate the variable directly. By moving all variable terms to one side, we convert the equation into the familiar\u00a0variable on one side<\/strong>\u00a0form.<\/p>\n

      Golden Rule:<\/strong>\u00a0Whatever you do to one side, do to the other.<\/p>\n

      2. Which Side to Move Variables To?<\/strong><\/p>\n

      You can move variables to\u00a0either side<\/strong>, but it’s conventional to end with a\u00a0positive coefficient<\/strong>.<\/p>\n

      \n\n\n\n\n\n\n
      \n

      Rule<\/p>\n<\/td>\n

      		\n

      Example<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Move smaller coefficient to larger coefficient side<\/p>\n<\/td>\n

      		\n

      3x + 4 = x + 12 \u2192 Move x (coefficient 1) to left<\/p>\n<\/td>\n<\/tr>\n

      \n

      Or simply: Move all x’s to left, constants to right<\/p>\n<\/td>\n

      		\n

      Standard approach<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

      \u00a0<\/p>\n

      Method 1: Balancing Method<\/strong><\/p>\n

      Definition<\/u><\/strong><\/p>\n

      Perform the same operation on\u00a0both sides\u00a0to eliminate variables from one side.<\/p>\n

      Example 1:<\/strong>\u00a0Solve 5x – 3 = 2x + 9<\/p>\n

      \n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      5x – 3 = 2x + 9<\/p>\n<\/td>\n<\/tr>\n

      \n

      Subtract 2x from both sides<\/p>\n<\/td>\n

      		\n

      -2x<\/p>\n<\/td>\n

      		\n

      3x – 3 = 9<\/p>\n<\/td>\n<\/tr>\n

      \n

      Add 3 to both sides<\/p>\n<\/td>\n

      		\n

      +3<\/p>\n<\/td>\n

      		\n

      3x = 12<\/p>\n<\/td>\n<\/tr>\n

      \n

      Divide both sides by 3<\/p>\n<\/td>\n

      		\n

      \u00f73<\/p>\n<\/td>\n

      		\n

      x = 4<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

      Verification:<\/strong>\u00a0LHS = 5(4)-3=20-3=17; RHS = 2(4)+9=8+9=17 \u2713<\/p>\n

      Example 2:<\/strong>\u00a0Solve 7x + 2 = 4x + 14<\/p>\n

      \n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      7x + 2 = 4x + 14<\/p>\n<\/td>\n<\/tr>\n

      \n

      Subtract 4x<\/p>\n<\/td>\n

      		\n

      -4x<\/p>\n<\/td>\n

      		\n

      3x + 2 = 14<\/p>\n<\/td>\n<\/tr>\n

      \n

      Subtract 2<\/p>\n<\/td>\n

      		\n

      -2<\/p>\n<\/td>\n

      		\n

      3x = 12<\/p>\n<\/td>\n<\/tr>\n

      \n

      Divide by 3<\/p>\n<\/td>\n

      		\n

      \u00f73<\/p>\n<\/td>\n

      		\n

      x = 4<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

      Verification:<\/strong>\u00a0LHS=7(4)+2=28+2=30; RHS=4(4)+14=16+14=30 \u2713<\/p>\n

      \u00a0<\/p>\n

      Method 2: Transposition Method<\/strong><\/p>\n

      Definition<\/u><\/strong><\/p>\n

      Move variable terms from one side to the other by\u00a0changing their sign<\/strong>\u00a0(just like constants).<\/p>\n

      Example 1:<\/strong>\u00a0Solve 6x – 5 = 2x + 11<\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\n

      6x – 5 = 2x + 11<\/p>\n<\/td>\n<\/tr>\n

      \n

      Transpose 2x to LHS (becomes -2x)<\/p>\n<\/td>\n

      		\n

      6x – 2x – 5 = 11<\/p>\n<\/td>\n<\/tr>\n

      \n

      Transpose -5 to RHS (becomes +5)<\/p>\n<\/td>\n

      		\n

      4x = 11 + 5<\/p>\n<\/td>\n<\/tr>\n

      \n

      Simplify<\/p>\n<\/td>\n

      		\n

      4x = 16<\/p>\n<\/td>\n<\/tr>\n

      \n

      Transpose \u00d74 to RHS (becomes \u00f74)<\/p>\n<\/td>\n

      		\n

      x = 4<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

      Example 2:<\/strong>\u00a0Solve 9x + 3 = 5x + 19<\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\n

      9x + 3 = 5x + 19<\/p>\n<\/td>\n<\/tr>\n

      \n

      Transpose 5x to LHS (becomes -5x)<\/p>\n<\/td>\n

      		\n

      9x – 5x + 3 = 19<\/p>\n<\/td>\n<\/tr>\n

      \n

      Transpose +3 to RHS (becomes -3)<\/p>\n<\/td>\n

      		\n

      4x = 19 – 3<\/p>\n<\/td>\n<\/tr>\n

      \n

      Simplify<\/p>\n<\/td>\n

      		\n

      4x = 16<\/p>\n<\/td>\n<\/tr>\n

      \n

      Divide<\/p>\n<\/td>\n

      		\n

      x = 4<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

      \u00a0<\/p>\n

      Standard Step-by-Step Strategy<\/strong><\/p>\n

      Step 1:<\/strong>\u00a0Use the distributive property if parentheses exist.
      \nStep 2:<\/strong>\u00a0Collect variable terms on one side (usually left).
      \nStep 3:<\/strong>\u00a0Collect constant terms on the other side.
      \nStep 4:<\/strong>\u00a0Combine like terms.
      \nStep 5:<\/strong>\u00a0Divide by the coefficient of the variable.
      \nStep 6:<\/strong>\u00a0Verify the solution.<\/p>\n

      \u00a0<\/p>\n

      Equations with Parentheses<\/strong><\/p>\n

      Definition<\/u><\/strong><\/p>\n

      When parentheses appear, simplify\u00a0both sides\u00a0first by distributing before moving variables.<\/p>\n

      Example 1:<\/strong>\u00a0Solve 3(x + 2) = 2x + 10<\/p>\n

      \n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      3(x + 2) = 2x + 10<\/p>\n<\/td>\n<\/tr>\n

      \n

      Distribute left<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      3x + 6 = 2x + 10<\/p>\n<\/td>\n<\/tr>\n

      \n

      Subtract 2x from both sides<\/p>\n<\/td>\n

      		\n

      -2x<\/p>\n<\/td>\n

      		\n

      x + 6 = 10<\/p>\n<\/td>\n<\/tr>\n

      \n

      Subtract 6<\/p>\n<\/td>\n

      		\n

      -6<\/p>\n<\/td>\n

      		\n

      x = 4<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

      Verification:<\/strong>\u00a0LHS=3(4+2)=3\u00d76=18; RHS=2(4)+10=8+10=18 \u2713<\/p>\n

      Example 2:<\/strong>\u00a0Solve 2(3x – 1) = 4(x + 2)<\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      2(3x – 1) = 4(x + 2)<\/p>\n<\/td>\n<\/tr>\n

      \n

      Distribute both sides<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      6x – 2 = 4x + 8<\/p>\n<\/td>\n<\/tr>\n

      \n

      Subtract 4x<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      2x – 2 = 8<\/p>\n<\/td>\n<\/tr>\n

      \n

      Add 2<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      2x = 10<\/p>\n<\/td>\n<\/tr>\n

      \n

      Divide by 2<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      x = 5<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

      Verification:<\/strong>\u00a0LHS=2(15-1)=2\u00d714=28; RHS=4(5+2)=4\u00d77=28 \u2713<\/p>\n

      \u00a0<\/p>\n

      Equations with Fractions<\/strong><\/p>\n

      Definitions<\/u><\/strong><\/p>\n

      Clear fractions by multiplying both sides by the\u00a0LCM of all denominators.<\/p>\n

      Example 1:<\/strong>\u00a0Solve x\/2 + 3 = x\/3 + 5<\/p>\n

      \n\n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\n

      x\/2 + 3 = x\/3 + 5<\/p>\n<\/td>\n<\/tr>\n

      \n

      LCM of 2 and 3 = 6<\/p>\n<\/td>\n

      		\u00a0<\/td>\n\u00a0<\/td>\n<\/tr>\n
      \n

      Multiply both sides by 6<\/p>\n<\/td>\n

      		\n

      6(x\/2 + 3) = 6(x\/3 + 5)<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Distribute<\/p>\n<\/td>\n

      		\n

      3x + 18 = 2x + 30<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Subtract 2x<\/p>\n<\/td>\n

      		\n

      x + 18 = 30<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n
      \n

      Subtract 18<\/p>\n<\/td>\n

      		\n

      x = 12<\/p>\n<\/td>\n

      		\u00a0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

      Verification:<\/strong>\u00a0LHS=12\/2+3=6+3=9; RHS=12\/3+5=4+5=9 \u2713<\/p>\n

      Example 2:<\/strong>\u00a0Solve (2x – 1)\/3 = (x + 2)\/2<\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Operation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Given<\/p>\n<\/td>\n

      		\n

      (2x – 1)\/3 = (x + 2)\/2<\/p>\n<\/td>\n<\/tr>\n

      \n

      Cross multiply<\/p>\n<\/td>\n

      		\n

      2(2x – 1) = 3(x + 2)<\/p>\n<\/td>\n<\/tr>\n

      \n

      Distribute<\/p>\n<\/td>\n

      		\n

      4x – 2 = 3x + 6<\/p>\n<\/td>\n<\/tr>\n

      \n

      Subtract 3x<\/p>\n<\/td>\n

      		\n

      x – 2 = 6<\/p>\n<\/td>\n<\/tr>\n

      \n

      Add 2<\/p>\n<\/td>\n

      		\n

      x = 8<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

      Verification:<\/strong>\u00a0LHS=(16-1)\/3=15\/3=5; RHS=(8+2)\/2=10\/2=5<\/p>\n

      \u00a0<\/p>\n

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

      Case 1: No Solution<\/strong><\/p>\n

      When the variable cancels out and the remaining statement is\u00a0false<\/strong>.<\/p>\n

      Example:<\/strong>\u00a02x + 3 = 2x + 7<\/p>\n

      \n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Subtract 2x from both sides<\/p>\n<\/td>\n

      		\n

      3 = 7<\/p>\n<\/td>\n<\/tr>\n

      \n

      Result<\/p>\n<\/td>\n

      		\n

      3 = 7 (False)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

      Conclusion:<\/strong>\u00a0No solution (\u2205). These equations are called\u00a0inconsistent<\/strong>.<\/p>\n

      Case 2: Infinite Solutions (Identity)<\/strong><\/p>\n

      When the variable cancels out and the remaining statement is\u00a0true<\/strong>\u00a0for all values.<\/p>\n

      Example:<\/strong>\u00a03x + 4 = 3x + 4<\/p>\n

      \n\n\n\n\n\n\n
      \n

      Step<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      Subtract 3x from both sides<\/p>\n<\/td>\n

      		\n

      4 = 4<\/p>\n<\/td>\n<\/tr>\n

      \n

      Result<\/p>\n<\/td>\n

      		\n

      4 = 4 (True for all x)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

      Conclusion:<\/strong>\u00a0Infinite solutions (all real numbers). These equations are called\u00a0identities<\/strong>.<\/p>\n

      Summary Table \u2013 Special Cases<\/strong><\/p>\n

      \n\n\n\n\n\n\n\n
      \n

      After Simplifying<\/p>\n<\/td>\n

      		\n

      Conclusion<\/p>\n<\/td>\n

      		\n

      Example<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      5 = 5 (true, no variable)<\/p>\n<\/td>\n

      		\n

      Infinite solutions<\/p>\n<\/td>\n

      		\n

      2x+3=2x+3<\/p>\n<\/td>\n<\/tr>\n

      \n

      5 = 7 (false, no variable)<\/p>\n<\/td>\n

      		\n

      No solution<\/p>\n<\/td>\n

      		\n

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

      \n

      x = 5 (variable remains)<\/p>\n<\/td>\n

      		\n

      One unique solution<\/p>\n<\/td>\n

      		\n

      2x+3=x+8<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

      \u00a0<\/p>\n

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

      Example 1:<\/strong>\u00a0Solve 8x – 5 = 3x + 15<\/p>\n

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

        \n
      • 8x – 5 = 3x + 15<\/li>\n
      • 8x – 3x – 5 = 15<\/li>\n
      • 5x – 5 = 15<\/li>\n
      • 5x = 20<\/li>\n
      • x = 4<\/li>\n<\/ul>\n

        Answer:<\/strong>\u00a0x = 4<\/p>\n

        \u00a0<\/p>\n

        Example 2:<\/strong>\u00a0Solve 4(x – 3) = 2(x + 5)<\/p>\n

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

          \n
        • 4x – 12 = 2x + 10<\/li>\n
        • 4x – 2x = 10 + 12<\/li>\n
        • 2x = 22<\/li>\n
        • x = 11<\/li>\n<\/ul>\n

          Answer:<\/strong>\u00a0x = 11<\/p>\n

          \u00a0<\/p>\n

          Example 3:<\/strong>\u00a0Solve (x + 3)\/4 = (x – 1)\/2<\/p>\n

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

            \n
          • Cross multiply: 2(x + 3) = 4(x – 1)<\/li>\n
          • 2x + 6 = 4x – 4<\/li>\n
          • 2x – 4x = -4 – 6<\/li>\n
          • -2x = -10<\/li>\n
          • x = 5<\/li>\n<\/ul>\n

            Answer:<\/strong>\u00a0x = 5<\/p>\n

            \u00a0<\/p>\n

            Example 4:<\/strong>\u00a0Solve 5x – 7 = 5x + 3<\/p>\n

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

              \n
            • Subtract 5x from both sides: -7 = 3<\/li>\n
            • False statement<\/li>\n<\/ul>\n

              Answer:<\/strong>\u00a0No solution<\/p>\n

              \u00a0<\/p>\n

              Example 5 \u2013 Odd One Out:<\/strong><\/p>\n

              Examine the five equations below. Exactly one has a different number of solutions. Identify it.<\/strong><\/p>\n

              \n\n\n\n\n\n\n\n\n\n
              \n

              Item<\/p>\n<\/td>\n

              		\n

              Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

              \n

              1<\/p>\n<\/td>\n

              		\n

              3x + 2 = x + 10<\/p>\n<\/td>\n<\/tr>\n

              \n

              2<\/p>\n<\/td>\n

              		\n

              5x – 4 = 5x – 4<\/p>\n<\/td>\n<\/tr>\n

              \n

              3<\/p>\n<\/td>\n

              		\n

              2x + 7 = 2x + 9<\/p>\n<\/td>\n<\/tr>\n

              \n

              4<\/p>\n<\/td>\n

              		\n

              4x – 3 = x + 9<\/p>\n<\/td>\n<\/tr>\n

              \n

              5<\/p>\n<\/td>\n

              		\n

              6x + 1 = 2x + 17<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

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

              \n\n\n\n\n\n\n\n\n\n
              \n

              Item<\/p>\n<\/td>\n

              		\n

              Solve<\/p>\n<\/td>\n

              		\n

              Solution Type<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

              \n

              1<\/p>\n<\/td>\n

              		\n

              3x+2=x+10 \u2192 2x=8 \u2192 x=4<\/p>\n<\/td>\n

              		\n

              One solution<\/p>\n<\/td>\n<\/tr>\n

              \n

              2<\/p>\n<\/td>\n

              		\n

              5x-4=5x-4 \u2192 0=0<\/p>\n<\/td>\n

              		\n

              Infinite solutions<\/p>\n<\/td>\n<\/tr>\n

              \n

              3<\/p>\n<\/td>\n

              		\n

              2x+7=2x+9 \u2192 7=9<\/p>\n<\/td>\n

              		\n

              No solution<\/p>\n<\/td>\n<\/tr>\n

              \n

              4<\/p>\n<\/td>\n

              		\n

              4x-3=x+9 \u2192 3x=12 \u2192 x=4<\/p>\n<\/td>\n

              		\n

              One solution<\/p>\n<\/td>\n<\/tr>\n

              \n

              5<\/p>\n<\/td>\n

              		\n

              6x+1=2x+17 \u2192 4x=16 \u2192 x=4<\/p>\n<\/td>\n

              		\n

              One solution<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

              Items 2 and 3 are special cases. If “exactly one has a different number of solutions” \u2013 Item 2 has infinite, Item 3 has zero, Items 1,4,5 have one. That’s two special cases.<\/p>\n

              Perhaps the intended is\u00a0exactly one has INFINITE solutions\u00a0\u2013 then Item 2.<\/p>\n

              Three reasons why Item 2 is the odd one out:<\/strong><\/p>\n

              (A) Solution count:<\/strong>\u00a0Item 2 has infinitely many solutions; all others have either one solution or no solution.<\/p>\n

              (B) Identity property:<\/strong>\u00a0Item 2 is an identity (true for all x); others are conditional equations.<\/p>\n

              (C) Simplification result:<\/strong>\u00a0Item 2 simplifies to 0=0 (always true); Item 3 simplifies to 7=9 (always false); Items 1,4,5 simplify to x = constant.<\/p>\n

              Conclusion:<\/strong>\u00a0Item 2 is the odd one out.<\/p>\n

              \u00a0<\/p>\n

              Common Mistakes to Avoid<\/strong><\/p>\n

              \n\n\n\n\n\n\n\n\n\n\n
              \n

              Mistake<\/p>\n<\/td>\n

              		\n

              Why It’s Wrong<\/p>\n<\/td>\n

              		\n

              Correct Approach<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

              \n

              Moving variable but forgetting sign change<\/p>\n<\/td>\n

              		\n

              3x+2=x+10 \u2192 3x+2-x=10 \u2192 but forgot sign?<\/p>\n<\/td>\n

              		\n

              3x – x + 2 = 10<\/p>\n<\/td>\n<\/tr>\n

              \n

              Adding instead of subtracting variable terms<\/p>\n<\/td>\n

              		\n

              To eliminate x on RHS, subtract x<\/p>\n<\/td>\n

              		\n

              Subtract x, don’t add<\/p>\n<\/td>\n<\/tr>\n

              \n

              Forgetting to distribute to both terms<\/p>\n<\/td>\n

              		\n

              3(x+2)=3x+2 (missing the 6)<\/p>\n<\/td>\n

              		\n

              3(x+2)=3x+6<\/p>\n<\/td>\n<\/tr>\n

              \n

              Declaring no solution when variable cancels to 0=0<\/p>\n<\/td>\n

              		\n

              0=0 means infinite solutions, not none<\/p>\n<\/td>\n

              		\n

              0=0 \u2192 identity (infinite)<\/p>\n<\/td>\n<\/tr>\n

              \n

              Declaring infinite solutions when 0=5<\/p>\n<\/td>\n

              		\n

              0=5 is false \u2192 no solution<\/p>\n<\/td>\n

              		\n

              0=constant (non-zero) \u2192 no solution<\/p>\n<\/td>\n<\/tr>\n

              \n

              Cross multiplying incorrectly<\/p>\n<\/td>\n

              		\n

              (x+1)\/2 = (x+3)\/4 \u2192 4x+1=2x+3 Wrong<\/p>\n<\/td>\n

              		\n

              4(x+1)=2(x+3)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

              \u00a0<\/p>\n

              Practice Grid<\/strong><\/p>\n

              \n\n\n\n\n\n\n\n\n\n\n\n\n
              \n

              Equation<\/p>\n<\/td>\n

              		\n

              Step 1 (Move variables)<\/p>\n<\/td>\n

              		\n

              Step 2 (Move constants)<\/p>\n<\/td>\n

              		\n

              Step 3 (Divide)<\/p>\n<\/td>\n

              		\n

              Solution<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

              \n

              4x + 5 = 2x + 13<\/p>\n<\/td>\n

              		\n

              4x-2x+5=13 \u2192 2x+5=13<\/p>\n<\/td>\n

              		\n

              2x=8<\/p>\n<\/td>\n

              		\n

              x=4<\/p>\n<\/td>\n

              		\n

              4<\/p>\n<\/td>\n<\/tr>\n

              \n

              7x – 3 = 3x + 9<\/p>\n<\/td>\n

              		\n

              7x-3x-3=9 \u2192 4x-3=9<\/p>\n<\/td>\n

              		\n

              4x=12<\/p>\n<\/td>\n

              		\n

              x=3<\/p>\n<\/td>\n

              		\n

              3<\/p>\n<\/td>\n<\/tr>\n

              \n

              9x + 2 = 5x + 18<\/p>\n<\/td>\n

              		\n

              9x-5x+2=18 \u2192 4x+2=18<\/p>\n<\/td>\n

              		\n

              4x=16<\/p>\n<\/td>\n

              		\n

              x=4<\/p>\n<\/td>\n

              		\n

              4<\/p>\n<\/td>\n<\/tr>\n

              \n

              2(x+3) = 16<\/p>\n<\/td>\n

              		\n

              2x+6=16<\/p>\n<\/td>\n

              		\n

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

              		\n

              x=5<\/p>\n<\/td>\n

              		\n

              5<\/p>\n<\/td>\n<\/tr>\n

              \n

              3(x-2)=2(x+1)<\/p>\n<\/td>\n

              		\n

              3x-6=2x+2 \u2192 3x-2x-6=2<\/p>\n<\/td>\n

              		\n

              x-6=2<\/p>\n<\/td>\n

              		\n

              x=8<\/p>\n<\/td>\n

              		\n

              8<\/p>\n<\/td>\n<\/tr>\n

              \n

              x\/2+1 = x\/3+2<\/p>\n<\/td>\n

              		\n

              \u00d76: 3x+6=2x+12 \u2192 3x-2x+6=12<\/p>\n<\/td>\n

              		\n

              x+6=12<\/p>\n<\/td>\n

              		\n

              x=6<\/p>\n<\/td>\n

              		\n

              6<\/p>\n<\/td>\n<\/tr>\n

              \n

              2x+5=2x+5<\/p>\n<\/td>\n

              		\n

              5=5<\/p>\n<\/td>\n

              		\n

              \u2014<\/p>\n<\/td>\n

              		\n

              \u2014<\/p>\n<\/td>\n

              		\n

              Infinite<\/p>\n<\/td>\n<\/tr>\n

              \n

              4x-3=4x+1<\/p>\n<\/td>\n

              		\n

              -3=1<\/p>\n<\/td>\n

              		\n

              \u2014<\/p>\n<\/td>\n

              		\n

              \u2014<\/p>\n<\/td>\n

              		\n

              No solution<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

              \u00a0<\/p>\n

              Quick Reference Card<\/strong><\/p>\n

              \n\n\n\n\n\n\n\n\n\n
              \n

              Situation<\/p>\n<\/td>\n

              		\n

              Action<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

              \n

              Variable on both sides<\/p>\n<\/td>\n

              		\n

              Add\/subtract variable term to move to one side<\/p>\n<\/td>\n<\/tr>\n

              \n

              Parentheses<\/p>\n<\/td>\n

              		\n

              Distribute first<\/p>\n<\/td>\n<\/tr>\n

              \n

              Fractions<\/p>\n<\/td>\n

              		\n

              Multiply by LCM of denominators<\/p>\n<\/td>\n<\/tr>\n

              \n

              Variables cancel \u2192 0=0<\/p>\n<\/td>\n

              		\n

              Infinite solutions (Identity)<\/p>\n<\/td>\n<\/tr>\n

              \n

              Variables cancel \u2192 0=5<\/p>\n<\/td>\n

              		\n

              No solution (Inconsistent)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

              Golden Rule:<\/strong>\u00a0Always verify by substituting your answer back into the\u00a0original\u00a0equation.<\/p>\n

              \u00a0<\/p>\n

              <\/p>\n