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Sep 17, 2022 · The above examples show us the following pattern: when there is one free variable in a consistent matrix equation, the solution set is a line, and when there are two free variables, the solution set is a plane, etc.
Jul 27, 2023 · The Geometry of Solution Sets: Hyperplanes; Particular Solution \(+\) Homogeneous solutions; Linearity and these parts. Contributor; Algebra problems can have multiple solutions. For example \(x(x-1)=0\) has two solutions: \(0\) and \(1\). By contrast, equations of the form \(Ax=b\) with \(A\) a linear operator have have the following property.
Nov 16, 2022 · Example 1 Show that each of the following numbers are solutions to the given equation or inequality. x = 3 x = 3 in x2 −9 = 0 x 2 − 9 = 0. y = 8 y = 8 in 3(y+1) =4y−5 3 ( y + 1) = 4 y − 5. z = 1 z = 1 in 2(z−5) ≤ 4z 2 ( z − 5) ≤ 4 z. z = −5 z = − 5 in 2(z−5) ≤ 4z 2 ( z − 5) ≤ 4 z.
The above examples show us the following pattern: when there is one free variable in a consistent matrix equation, the solution set is a line, and when there are two free variables, the solution set is a plane, etc.
Solution set. In mathematics, a solution set is the set of values that satisfy a given set of equations or inequalities . For example, for a set of polynomials over a ring , the solution set is the subset of on which the polynomials all vanish (evaluate to 0), formally.
A solution set is an ordered triple {(x,y,z)} that represents the intersection of three planes in space. See Example \(\PageIndex{1}\). A system of three equations in three variables can be solved by using a series of steps that forces a variable to be eliminated.
One can describe a solution set in any of the following ways: In Words: a 2 = 25 has solutions 5 and -5. (That is, a 2 = 25 is true when a = 5 or a = -5.) In Set Notation: The solution set of a 2 = 25 is {5, -5}. It is awkward to express the set of infinitely many numbers in set notation.
