{"id":23201,"date":"2020-10-29T19:32:25","date_gmt":"2020-10-29T14:02:25","guid":{"rendered":"https:\/\/e-gmat.com\/blogs\/?p=23201"},"modified":"2026-01-24T21:27:33","modified_gmt":"2026-01-24T15:57:33","slug":"introduction-to-gmat-algebra-algebraic-expressions-linear-quadratic-equations","status":"publish","type":"post","link":"https:\/\/e-gmat.com\/blogs\/introduction-to-gmat-algebra-algebraic-expressions-linear-quadratic-equations\/","title":{"rendered":"GMAT Algebra and Algebraic Expressions | Linear & Quadratic Equations"},"content":{"rendered":"A <\/span> 9<\/span> min read <\/span><\/span>

The word Algebra was derived from the Arabic word \u201cal-jabr\u201d <\/em>meaning \u201creunion of broken parts\u201d. \u00a0In algebra, arithmetic computations are carried out with letters standing for numbers. Many students find algebra difficult and through this article, we aim to demystify GMAT Algebra and how you can solve GMAT Algebra questions with ease.<\/p>\n

\u00a0<\/p>\n

\"Introduction<\/p>\n

Simple Algebraic expression<\/h2>\n

The basics of algebra start with understanding simple algebraic expressions. In this section, we will learn more about the different terms & the terminologies used in algebraic expressions.<\/p>\n

Algebraic expression:<\/h3>\n

An algebraic expression is made up of:<\/p>\n

    \n
  • Constants, such as 2, 3, 4, 5, etc,<\/li>\n
  • Variables, such as (a, b, c, d, \u2026, x, y, x) and,<\/li>\n
  • Algebraic operations, such as addition, subtraction, multiplication, etc.<\/li>\n<\/ul>\n

    Simple examples of algebraic expression are: 2x + 3, 3y + 2, x\u00b2\u00a0+ 3 etc.<\/p>\n

    Terminologies used in an algebraic expression\"linear<\/h3>\n

    An algebraic expression includes:<\/p>\n

    1. Terms<\/h4>\n

    For example: In the expression 9x + 4y + 2<\/p>\n

      \n
    • 9x, 4y, and 2 are three terms.<\/li>\n<\/ul>\n

      2. Variable<\/h4>\n

      x and y are variables in the equation 9x + 4y + 2 as their value is not defined.<\/p>\n

      3. Constant<\/h4>\n

      2 is a constant term in the equation 9x + 4y + 2 as it has one unique and defined value.<\/p>\n

      4. Coefficient of a variable<\/h4>\n

      The constant attached to the variable<\/u><\/em> is known as the coefficient of the variable.<\/p>\n

        \n
      • In the expression 9x + 4y + 2,\n
          \n
        • Coefficient of x = 9<\/li>\n
        • Coefficient of y = 4<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

          5. Degree<\/h4>\n

          In algebra, the highest power of the variable<\/u><\/em> is known as the degree of the expression.<\/p>\n

          For example:<\/p>\n\n\n\n\n\n\n
          Expression<\/strong><\/td>\nDegree<\/strong><\/td>\n<\/tr>\n
          x\u00b2\u00a0+ 2<\/td>\n2 (Power of x)<\/td>\n<\/tr>\n
          x + y\u00b3\u00a0+ 2<\/td>\n3 (Power of y)<\/td>\n<\/tr>\n
          3x + 4y<\/td>\n1 (Power of x or y)<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

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

            \n
          1. If the degree of an algebraic expression is 1 then it is called linear expression.<\/em><\/li>\n
          2. If the degree of an algebraic expression is 2 then it is called quadratic expression.<\/em><\/li>\n<\/ol>\n

            Algebraic Equation<\/h2>\n

            When an expression is equal to a constant\/ another expression<\/u><\/em> then an equation is obtained.<\/p>\n

            For example: 5x and 3x + 4 are two different expressions.<\/p>\n

              \n
            • While 5x = 3x + 4 is one equation.<\/li>\n
            • Similarly, 4y + 5 = 2 is also an equation.<\/li>\n<\/ul>\n

              The solution of an equation<\/h3>\n

              The solution of an equation is the value of the variable for which expression on the left-hand side is equal to the expression on the right-hand side.<\/p>\n

              For example:<\/strong><\/p>\n

                \n
              • For y = \u22123\/4 , 4y + 5 is equal to 2\n
                  \n
                • How we arrived at y = \u22123\/4 \u00a0will be discussed later in this article.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                  Linear Equations<\/h2>\n

                  \"Linear<\/strong><\/p>\n

                  Algebraic equations of degree one are called linear equations.<\/p>\n

                  For example: 3x + 9 = 0, a + 2b + 3c = 0 etc.<\/p>\n

                  A linear equation can be either in one variable or more than one variable.<\/p>\n

                    \n
                  • 3x + 9 = 0 is a linear equation in one variable.<\/li>\n
                  • And, a + 2b + 3c = 0 is a linear equation in three variables.<\/li>\n<\/ul>\n

                    The most common linear equation involves either one variable or two variables.<\/p>\n

                    Let us learn how to solve both types of linear equations.<\/p>\n

                    \u00a0Linear equation in one variable<\/h3>\n

                    The standard form of linear equation in one variable is ax + b = 0<\/em> where a \u2260 0.<\/p>\n

                      \n
                    • And, it is very easy to find the solution to this type of linear equations.<\/li>\n<\/ul>\n

                      Solving a Linear equation in one variable<\/h4>\n

                      To solve a linear equation of one variable:<\/p>\n

                        \n
                      • We can move the constant terms to one side and variable terms to another side of the equality sign.<\/li>\n
                      • For example: 2x + 3 = 2 can also be written as:\n
                          \n
                        • 2x = 2 \u2212\u00a03 by moving 3 to the other side of the equality sign.<\/li>\n
                        • Hence, 2x = \u22122<\/li>\n
                        • x = \u22121<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                          That is how we can easily solve the linear equations in one variable.<\/p>\n

                          Linear equation in two variables<\/h3>\n

                          The standard form of linear equation in two variables is: ax + by + c = 0<\/em> where a, b \u2260 0.<\/p>\n

                          For example:<\/p>\n

                            \n
                          • x + 2y = 3 Or 3x + 4y = 6 etc.<\/li>\n
                          • Both the equation involves two variables: x and y.<\/li>\n<\/ul>\n

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

                            To solve linear equation in two variables, we need two distinct equations<\/strong> to find the value of both the variables.<\/em><\/p>\n

                            Let us understand how we can solve these types of equations.<\/p>\n

                            Solving a Linear equation in two variables<\/h4>\n

                            Linear equation in two variables can be solved by two methods:<\/p>\n

                              \n
                            1. Substitution Method<\/li>\n
                            2. Elimination method<\/li>\n<\/ol>\n
                              Substitution Method<\/h5>\n

                              As the name suggests, in this method:<\/p>\n

                                \n
                              1. We find the value of one of the variables in terms of another variable from one of the equations.<\/li>\n
                              2. Then, we substitute the value of that variable in another equation and get an equation in one variable only.\n
                                  \n
                                1. From this equation, we can find the value of one variable.<\/li>\n
                                2. Now, substitute this value in any of the equations to get the value of another variable.<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n

                                  For example:<\/strong><\/p>\n

                                  We have two equations: x + 3y = 4 and x + y = 2.<\/p>\n

                                    \n
                                  • We will find the value of either x in terms of y or y in terms of x from one equation.\n
                                      \n
                                    • We will use x + y = 2 to get x = 2 \u2212 y<\/li>\n<\/ul>\n<\/li>\n
                                    • Let us now substitute this value of x in another equation i.e. x + 3y\u00a0 = 4\n
                                        \n
                                      • (2 \u2212 y) + 3y = 4<\/li>\n
                                      • 2 + 2y = 4<\/li>\n
                                      • y = 1<\/li>\n<\/ul>\n<\/li>\n
                                      • Substituting y = 1 in x + y = 2, we get x = 1.<\/li>\n<\/ul>\n

                                        That is how we use the substitution method.<\/p>\n

                                        \u00a0<\/p>\n

                                        Elimination Method<\/h5>\n

                                        In this method, we first eliminate one of the variables by using algebraic manipulation such as addition, subtraction, multiplication, etc to get the equation in one variable form.<\/p>\n

                                          \n
                                        • And, once we have the value of one variable, we can easily find the value of another variable.<\/li>\n<\/ul>\n

                                          Let us take the example of<\/p>\n

                                            \n
                                          • x + 3y = 4 ——— Equation 1<\/strong><\/li>\n
                                          • and x + y = 2 ——– Equation 2<\/strong><\/li>\n<\/ul>\n

                                            and solve it by the elimination method.<\/p>\n

                                            We can eliminate either x or y.<\/p>\n

                                            By eliminating y<\/u><\/em><\/p>\n

                                              \n
                                            • As the coefficient of y is 3 in equation 1 and 1 in equation 2, we can multiply equation 2 by 3 so that the coefficient of y becomes equal in both equations.\n
                                                \n
                                              • Multiplying equation 2 by 3, we get:\n
                                                  \n
                                                1. 3x + 3y = 6 ——– Equation 3<\/strong><\/li>\n<\/ol>\n<\/li>\n
                                                2. On subtracting equation 1 from equation 2, we get\n
                                                    \n
                                                  1. (3x + 3y) \u2212 (x + 3y) = 6 – 4<\/li>\n
                                                  2. 2x = 2 or x = 1<\/li>\n<\/ol>\n<\/li>\n
                                                  3. Substituting x = 1 in x + y = 2, we get y = 1.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                                    By eliminating x<\/u><\/em><\/p>\n

                                                      \n
                                                    • As the coefficient of x is 1 in both the equations, it can be eliminated by subtracting both the equations.<\/li>\n
                                                    • After completing the process, we will get x = 1 and y = 1.<\/li>\n<\/ul>\n

                                                      That is how we use the substitution method.<\/p>\n

                                                      So, we have learnt how to solve linear equations in one variable and two variables.<\/p>\n

                                                      Let us now understand another type of algebraic equation i.e. Quadratic Equation.<\/p>\n

                                                      \u00a0<\/p>\n

                                                      Quadratic Equations<\/h2>\n

                                                      Algebraic equations of degree two are quadratic equations.<\/p>\n

                                                      For example: x\u00b2\u00a0+ 9 = 0, y\u00b2\u00a0+ 3y + 2 = 0 etc.<\/p>\n

                                                      The standard form of a quadratic equation is: ax\u00b2\u00a0+ bx + c = 0 where a \u2260 0.<\/p>\n

                                                      For example:<\/p>\n

                                                      1. For x\u00b2\u00a0+ 3x + 5 = 0,<\/p>\n

                                                      a) a = 1, b = 3 and c = 5.<\/p>\n

                                                      2. For y\u00b2\u00a0+ 9y + 8 = 1<\/p>\n

                                                      a) a = 1, b = 9 and c = 7.<\/p>\n

                                                      b) c is not 8 as y\u00b2\u00a0+ 9y + 8 = 1 is not in the standard form of ax\u00b2\u00a0+ bx +c = 0.<\/u><\/em><\/p>\n

                                                      (i) The standard form is y\u00b2\u00a0+ 9y + 7 = 0 (RHS must be 0)<\/strong><\/p>\n

                                                      Root of a quadratic equation<\/h3>\n

                                                      Roots of a quadratic equation are nothing but the solutions of the quadratic equation.<\/p>\n

                                                        \n
                                                      • So, the value of a variable that satisfies a quadratic equation is a root of the quadratic equation.<\/u><\/em><\/li>\n<\/ul>\n

                                                        A quadratic equation can have a maximum of two real roots.<\/p>\n

                                                          \n
                                                        • It can have 0 or 1 or 2 roots.<\/li>\n<\/ul>\n

                                                          We can find the roots of a quadratic equation by two methods:<\/p>\n

                                                            \n
                                                          1. Factor Method<\/li>\n
                                                          2. Quadratic Formula<\/li>\n<\/ol>\n

                                                            Factor method<\/h4>\n

                                                            To factor a quadratic equation, ax\u00b2\u00a0+ bx + c = 0:<\/p>\n

                                                            Step 1)-<\/strong> We need to take two numbers such that their sum\/difference is b and product is ac.<\/em><\/p>\n

                                                              \n
                                                            • For example: For x\u00b2\u00a0+ 4x + 3 two such numbers are 3 and 1 as 3 + 1 = 4 and 3 \u00d7 1 = 3.<\/li>\n<\/ul>\n

                                                              Step 2)- <\/strong>Write the middle term as the sum or difference of the two numbers.<\/p>\n

                                                                \n
                                                              • We can write 4x as 3x + x.<\/li>\n
                                                              • x\u00b2\u00a0+ 4x + 3 = x\u00b2\u00a0+ 3x + x + 3<\/li>\n<\/ul>\n

                                                                Step 3)- <\/strong>Factor the first two terms and last two terms.<\/p>\n

                                                                  \n
                                                                • First two terms are: x\u00b2\u00a0+ 3x. This is factored as x (x + 3)<\/li>\n
                                                                • Last two terms are: x + 3. This is factored as 1 (x +3)\n
                                                                    \n
                                                                  • Hence, x\u00b2\u00a0+ 3x + x + 3 = x (x + 3) + 1 (x +3)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                                                    Step 4)- <\/strong>Now both the terms should have a common factor. Take that common factor out.<\/p>\n

                                                                      \n
                                                                    • (x+3) is common in x (x + 3) and 1 (x + 3)\n
                                                                        \n
                                                                      • Hence, x (x + 3) + 1 (x + 3) = (x + 3) (x + 1)<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n

                                                                        So, this complete process is the factor method.<\/p>\n

                                                                        Hence, the quadratic equation becomes (x + 3) (x + 1) = 0<\/p>\n

                                                                        Now, the product of (x + 3) and (x + 1) = 0 can be 0 only in two cases:<\/p>\n

                                                                        Case-1) <\/strong>When x + 1 is 0<\/p>\n

                                                                          \n
                                                                        • x = \u22121<\/li>\n<\/ul>\n

                                                                          Case-2) <\/strong>When x + 3 is 0<\/p>\n

                                                                            \n
                                                                          • x = \u22123.<\/li>\n<\/ul>\n

                                                                            Hence, \u22121 and \u22123 are the roots of x\u00b2\u00a0+ 4x +3.<\/p>\n

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

                                                                            Both the terms cannot be 0 because x + 3 is definitely greater than x + 1.<\/em><\/p>\n

                                                                            This is the complete process of factoring method.<\/p>\n

                                                                              \n
                                                                            • This method may seem a bit lengthy but the step that takes time is step 1 and once it is done remaining steps are a cakewalk.<\/em><\/li>\n<\/ul>\n

                                                                              Quadratic formula<\/h4>\n

                                                                              \"Quadratic<\/p>\n

                                                                              The root of any quadratic equation can be easily obtained by applying the formula:<\/p>\n

                                                                                \n
                                                                              • x= \u2212b \u00b1 \u221a (b\u00b2 \u2212 4ac) \/ 2a, where \u2212b + \u221a (b\u00b2 \u2212 4ac) \/ 2a is one root and \u2212b \u2212 \u221a (b\u00b2 \u2212 4ac) \/ 2a is another root.<\/li>\n<\/ul>\n

                                                                                For example: Let us try finding the roots of the equation x\u00b2\u00a0+ 4x +3 by using the quadratic formula.<\/p>\n

                                                                                  \n
                                                                                • In x\u00b2\u00a0+ 4x + 3, a = 1, b = 4, and c = 3<\/li>\n
                                                                                • So, roots = \u22124 \u00b1 \u221a (4\u00b2\u00a0\u2212 4 \u00d7 1 \u00d7 3) \/ 2 \u00d7 1\n
                                                                                    \n
                                                                                  • x = (\u22124\u00a0\u00b1 2) \/ 2 = \u22122 \u00b1 1<\/li>\n
                                                                                  • Hence, \u22121 and \u22123 are two roots of the equation.<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n\n
                                                                                    \n\n\n

                                                                                    Concepts Down. Now Build Your Quant Plan.<\/h2>\n\n\n\n

                                                                                    You have learned the algebra fundamentals. But mastering GMAT Quant requires systematic practice across all topics. Get a personalized plan that identifies your gaps and builds your skills step by step.<\/p>\n\n\n<\/div>\n\n\n\n\n\n