But it is not impossible that an equation cannot have more than one solution or an infinite number of solutions or no solutions at all. As far as we look there is usually one solution to an equation. For example, 2x + 4y - 9 where x and y are variables and 9 is a constant. An expression consists of variables like x or y and constant terms which are conjoined together using algebraic operators. An equation is an expression that has an equal sign (=) in between. Let's just quickly refresh the meanings of the terms once again before we dig in. We all are well acquainted with equations and expressions. Whereas the Infinite Solutions - Definition, Conditions, and Examples article prepared by Vedantu takes care of these elements and tends to make the article more and more interesting for the students. The reasons for the same could be the tough language of the content, lengthy and monotonous flow of the article and others. It is often seen, students are unable to comprehend the topic despite working hard over it. These teachers have a good deal of experience in the field of education and are very well aware of the needs of the students. Sometimes, a homogeneous system has non-zero vectors also to be solutions, To find them, we have to use the matrices and the elementary row operations.Infinite solutions article is developed by the highly qualified teachers of Vedantu. For example, (x, y) = (0, 0) is a solution of the homogeneous system x + y = 0, 2x - y = 0. What is the Solution of Homogeneous System of Linear Equations?Ī zero vector is always a solution to any homogeneous system of linear equations. ![]() If each equation in it has its constant term to be zero, then the system is said to be homogeneous. How do You Know if a System of Equations is Homogeneous?Ī system has two or more equations in it. Any other solution than the trivial solution (if any) is called a nontrivial solution. What are Trivial and Nontrivial Solutions of a Homogeneous System of Linear Equations?Ī vector formed by all zeros (zero vector) is always a solution of any homogeneous linear system and it is called a trivial solution. Now let us the expand the first two rows as equations:Īnswer: The solution is (x, y, z) = (-t, -2t, t), where 't' is a real number.įAQs on Homogeneous System of Linear Equations What is a Homogeneous Linear Equation Example?Ī homogeneous linear equation is a linear equation in which the constant term is 0. Let us find them using the elementary row operations on the coefficient matrix.ĭividing the 2 nd row by 51 and and 3 rd row by 17, Therefore, the system has an infinite number of solutions (along with the trivial solution (x, y, z) = (0, 0, 0)). Let us find the determinant of the coefficient matrix: When t = 0.5: (x, y, z) = (-1, 0.5, 0.5), etcĮxample 3: How many solutions does the following system has? Find them all. For example, some nontrivial solutions of the above homogeneous system can be: Thus, the solution is (x, y, z) = (-2t, t, t) which represents an infinite number of nontrivial solutions as 't' can be replaced with one of the real numbers (which is an infinite set). Hence we should assume one of the variables to be a parameter (say t which is a real number). We have two equations in three variables. ![]() Just expand the first two rows of the above matrix as equations. It means that the system has nontrivial solutions also. We couldn't convert it into the upper diagonal matrix as we ended up with a row of zeros in the matrix. Let us take the coefficient matrix of the above system and apply row operations in order to convert it into an upper diagonal matrix. We can find them using the matrix method and applying row operations. But it may (or may not) have other solutions than the trivial solutions that are called nontrivial solutions. For example, the system formed by three equations x + y + z = 0, y - z = 0, and x + 2y = 0 has the trivial solution (x, y, z) = (0, 0, 0). , 0) is obviously a solution to the system and is called the trivial solution (the most obvious solution). Since there is no constant term present in the homogeneous systems, (x₁, x₂. Solving Homogeneous System of Linear EquationsĪ homogeneous system may have two types of solutions: trivial solutions and nontrivial solutions.
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