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When seeking a solution, one or more variables are designated as unknowns. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality.
Sometimes boxes or shapes are used to indicate unknown values. As such, number sentences are used to introduce students to notions of structure and elementary algebra prior to a more formal treatment of these concepts. A number sentence without unknowns is equivalent to a logical proposition expressed using the notation of arithmetic.
Solving an equation containing variables consists of determining which values of the variables make the equality true. The variables for which the equation has to be solved are also called unknowns, and the values of the unknowns that satisfy the equality are called solutions of the equation.
To solve this kind of equation, the technique is add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable. [37] This problem and its solution are as follows: Solving for x
Consider a linear non-homogeneous ordinary differential equation of the form = + (+) = where () denotes the i-th derivative of , and denotes a function of .. The method of undetermined coefficients provides a straightforward method of obtaining the solution to this ODE when two criteria are met: [2]
Consider a system of n linear equations for n unknowns, represented in matrix multiplication form as follows: = where the n × n matrix A has a nonzero determinant, and the vector = (, …,) is the column vector of the variables. Then the theorem states that in this case the system has a unique solution, whose individual values for the unknowns ...