{"id":10298,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=10298"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"solution-techniques-substitution-elimination-and-standard-rules","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/solution-techniques-substitution-elimination-and-standard-rules\/","title":{"rendered":"Solution Techniques: Substitution, Elimination And Standard Rules"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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

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

Unit: Systems of Two Linear Equations with Two Variables<\/strong><\/h2>\n

Solution Techniques: Substitution, Elimination And Standard Rules<\/strong><\/h2>\n

Overview<\/p>\n

A system of two linear equations with two variables is a set of equations where each equation is linear and involves two variables, typically x<\/em> and y<\/em>. The general form of such a system is:<\/p>\n

\"\"<\/p>\n

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

where a<\/em>1<\/sub>\u200b, b<\/em>1<\/sub>\u200b<\/sub>, <\/sub>c<\/em>1<\/sub>\u200b, a<\/em>2<\/sub>\u200b, b<\/em>2<\/sub>\u200b, and c<\/sub><\/em>2<\/sub>\u200b are constants.<\/p>\n

Solutions to the System<\/strong><\/p>\n

The solution to a system of two linear equations can be:<\/p>\n

    \n
  1. One unique solution: The lines intersect at a single point.<\/li>\n
  2. No solution: The lines are parallel and do not intersect.<\/li>\n
  3. Infinitely many solutions: The lines coincide (are the same line).<\/li>\n<\/ol>\n

    Methods of Solving<\/strong><\/p>\n

      \n
    1. Graphical Method:<\/strong>\n
        \n
      • Plot both equations on the same set of axes.<\/li>\n
      • The point of intersection, if any, is the solution.<\/li>\n
      • If the lines are parallel, there is no solution.<\/li>\n
      • If the lines coincide, there are infinitely many solutions.<\/li>\n<\/ul>\n<\/li>\n
      • Substitution Method:<\/strong>\n
          \n
        • Solve one of the equations for one variable in terms of the other.<\/li>\n
        • Substitute this expression into the other equation.<\/li>\n
        • Solve the resulting single-variable equation.<\/li>\n
        • Substitute back to find the other variable.<\/li>\n<\/ul>\n<\/li>\n
        • Elimination (Addition) Method:<\/strong>\n
            \n
          • Multiply one or both equations by suitable constants to align coefficients.<\/li>\n
          • Add or subtract the equations to eliminate one variable.<\/li>\n
          • Solve the resulting single-variable equation.<\/li>\n
          • Substitute back to find the other variable.<\/li>\n<\/ul>\n<\/li>\n
          • Matrix Method (using Determinants):<\/strong>\n
              \n
            • Represent the system as a matrix equation AX<\/em>=B<\/em>.<\/li>\n
            • Use the inverse of the coefficient matrix A<\/em> to solve for X<\/em>, if A<\/em> is invertible.<\/li>\n
            • X<\/em>=A<\/em>\u22121B<\/em>.<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

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

              Consider the system:<\/p>\n

              \"\"<\/p>\n

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

              Graphical Method:<\/p>\n

                \n
              • Convert to slope-intercept form (if necessary) and plot the lines.<\/li>\n<\/ul>\n

                \"\"<\/p>\n

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

                  \n
                • Determine the intersection point.<\/strong><\/li>\n<\/ul>\n

                  Substitution Method:<\/p>\n

                    \n
                  • Solve the second equation for y<\/em>:\n
                      \n
                    • y<\/em>=4x<\/em>\u22125<\/li>\n<\/ul>\n<\/li>\n
                    • Substitute into the first equation:\n
                        \n
                      • 2x<\/em>+3(4x<\/em>\u22125)=6<\/li>\n
                      • 2x<\/em>+12x<\/em>\u221215=6<\/li>\n
                      • 14x<\/em>=21<\/li>\n
                      • x<\/em>=\n
                        \"\"<\/p>\n

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

                        =1.5<\/li>\n<\/ul>\n<\/li>\n

                      • Substitute x<\/em>=1.5 back into y<\/em>=4x<\/em>\u22125:\n
                          \n
                        • \ud835\udc66=4(1.5)\u22125=6\u22125=1<\/li>\n<\/ul>\n<\/li>\n
                        • Solution: (1.5,1)<\/li>\n<\/ul>\n

                          Elimination Method:<\/strong><\/p>\n

                            \n
                          • Align coefficients for elimination:\n
                              \n
                            • Multiply the first equation by 1 and the second by 3:<\/li>\n
                            • 2x<\/em>+3y<\/em>=6<\/li>\n
                            • 12x<\/em>\u22123y<\/em>=15<\/li>\n<\/ul>\n<\/li>\n
                            • Add the equations:\n
                                \n
                              • 14x<\/em>=21<\/li>\n
                              • x<\/em>=1.5<\/li>\n<\/ul>\n<\/li>\n
                              • Substitute x<\/em>=1.5 into the first equation to find \ud835\udc66y<\/em>:\n
                                  \n
                                • 2(1.5)+3y<\/em>=6<\/li>\n
                                • 3+3y<\/em>=6<\/li>\n
                                • 3y<\/em>=3<\/li>\n
                                • y<\/em>=1<\/li>\n<\/ul>\n<\/li>\n
                                • Solution: (1.5,1)<\/li>\n<\/ul>\n

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

                                    \n
                                  1. Parallel Lines:<\/strong>\n
                                      \n
                                    • The system<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                                      \"\"<\/p>\n

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

                                      \u00a0has no solution since the lines have the same slope but different y-intercepts.<\/p>\n

                                        \n
                                      1. Coincident Lines:<\/strong>\n
                                          \n
                                        • The system<\/li>\n<\/ul>\n<\/li>\n<\/ol>\n

                                          \"\"<\/p>\n

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

                                          has infinitely many solutions since the second equation is a multiple of the first.<\/p>\n

                                          Determinant Method (Cramer’s Rule)<\/strong><\/p>\n

                                          Cramer’s Rule is a mathematical theorem used to solve systems of linear equations with as many equations as unknowns, provided the system’s determinant is non-zero. Here, we’ll focus on using Cramer’s Rule for systems of two linear equations with two variables.<\/p>\n

                                          System of Equations<\/strong><\/p>\n

                                          Consider the system:<\/p>\n

                                          \"\"<\/p>\n

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

                                          Determinants<\/strong><\/p>\n

                                          To use Cramer’s Rule, we need to calculate three determinants:<\/p>\n

                                            \n
                                          1. Determinant D<\/em>: This is the determinant of the coefficient matrix.<\/li>\n
                                          2. Determinant Dx<\/em>\u200b: This is the determinant of the matrix obtained by replacing the x-coefficients with the constants from the right-hand side of the equations.<\/li>\n
                                          3. Determinant Dy<\/em>\u200b: This is the determinant of the matrix obtained by replacing the y-coefficients with the constants from the right-hand side of the equations.<\/li>\n<\/ol>\n

                                            Steps to Apply Cramer’s Rule<\/p>\n

                                              \n
                                            1. Determinant D<\/em>:<\/li>\n<\/ol>\n

                                              \"\"<\/p>\n

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

                                                \n
                                              1. Determinant Dx<\/em>\u200b:<\/li>\n<\/ol>\n

                                                \"\"<\/p>\n

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

                                                  \n
                                                1. Determinant Dy<\/em>\u200b:<\/li>\n<\/ol>\n

                                                  \"\"<\/p>\n

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

                                                    \n
                                                  1. Solve for \ud835\udc65x<\/em> and \ud835\udc66y<\/em>:<\/li>\n<\/ol>\n

                                                    \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 <\/p>\n

                                                    \"\"<\/p>\n

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

                                                    Provided \ud835\udc37\u22600, the system has a unique solution.<\/p>\n

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

                                                    One unique solution: Lines intersect at one point.<\/p>\n

                                                    No solution: Lines are parallel.<\/p>\n

                                                    Infinitely many solutions: Lines coincide.<\/p>\n

                                                    Methods include graphical, substitution, elimination, and matrix methods.<\/p>\n

                                                    Special cases highlight the nature of solutions based on the relationship between the lines.<\/p>\n

                                                    Understanding these fundamentals enables solving and analyzing systems of linear equations in various contexts.<\/p>\n

                                                    Cramer’s Rule provides a straightforward method to solve a system of linear equations using determinants, provided the determinant of the coefficient matrix is non-zero. The steps involve calculating the determinant of the coefficient matrix and two modified matrices, then using these determinants to find the values of the variables.<\/p>\n

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