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Q: Does every system of a linear equation have a solution?

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there is no linear equations that has no solution every problem has a solution

This is a linear equation in two variables and the coordinates of each and every point on the line that it describes is a solution. A single linear equation does not have an "answer".

Linear means referring to a line, so a line must be linear! Every point on any line is a solution to the equation that defines the line.

Every equation has a solution.

An equality symbol such as =

It's a linear equation in two variables . . . 'g' and 'p'. The graph of this equation is a straight line. The coordinates of every point on the line are a solution of the equation. There are an infinite number of them.

You are trying to find a set of values such that, if those values are substituted for the variables, every equation in the system is true.

Infinitely many. Assuming the variables are "x" and "y" , for every value of "x" a value for "y" can be calculated.

Yes, you can say something like y < infinity and y > -infinity .

Put the coefficients of the variables into a 3x3 matrix, and take the determinant of the matrix. If the determinant is not zero, then there is one solution. If the determinant is zero, then there are infinite solutions or there is no solution. Think of a system of 2 variables, for simplicity. You have 2 equations and 2 variables (x & y). Each equation can be graphed as a straight line, hence the name 'linear system'. If the 2 lines are not parallel, then there will be only one point where they intersect, which is the one solution to the linear system. If they are parallel, then there is no solution(they never intersect), and if the two lines coincide, then infinite solutions(they intersect at every point). In both of the latter cases, the related matrix will have a determinant of zero.

Equations with the same solution are called dependent equations, which are equations that represent the same line; therefore every point on the line of a dependent equation represents a solution. Since there is an infinite number of points on a line, there is an infinite number of simultaneous solutions. For example, 2x + y = 8 4x + 2y = 16 These equations are dependent. Since they represent the same line, all points that satisfy either of the equations are solutions of the system. A system of linear equations is consistent if there is only one solution for the system. A system of linear equations is inconsistent if it does not have any solutions.

Each and every value of y will satisfy any linear equation in which both x and y appear.

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