Linear equations take the form y= mx+b wherem is the slope [rise(y)/run(x) on a graph]x is the x-value any point on the graphy is the y-value of any point on the graphb is the y-intercept on the graphLinear equations take the form a1x1 + a2x2 + a3x3 + ... + anxn + an+1 = 0Each ai represents a constant, xi a variable.The equation above is linear in n dimensions. In two dimensions, linear equations are typically written ax + by = c. In three dimensions, ax + by + cz = d. After that, the form given above (with the subscripts) is preferred.Any system of n equations and n unknowns may have a unique solution. If two of the equations are multiples of each other, the solution set will not be unique, but represent a line, plane, or subspace. It is also possible the system may have no solution, such as the following:5x + 10y = 55x + 10y = 20This system represents two parallel lines--there is no solution.
You don't need ANY factor. To find a unique solution, or a few, you would usually need to have as many equations as you have variables.
The answer depends on the level of your knowledge. The High level, simple answer is first. The Low level slog follows:HIGH LEVEL, SIMPLESuppose you have n equations of the forma11x1 + a12x2 + ... + a1nxn = bn wherethe as are coefficients,x1, x2, ... xn are the unknown variablesandb1, b2, ... bn are the constants.Write the n linear equations in n unknowns in the form Ax= bwhereA is an n*n matrix of coefficientsx is the n*1 matrix of the unknown variablesandb is the n*1 matrix of the constants.Find the inverse of A.Then x = A-1b.The above method works if the system has a unique solution. If the n equations are not independent, you will need to use a generalised inverse and that starts to get rather complicated. If they are inconsistent, then neither the inverse nor generalised inverse will be found.LOW LEVEL SLOGUse the first equation to express x1 in terms of the other variables. Substitute this value for x1 in the remaining n-1 equations. You now have n-1 equations in n-1 unknown variables.Use the first of the new equations to express x2 in terms of the other variables. Substitute in remaining equations. You now have n-2 equations in n-2 unknown variables.Continue until you have 1 equation in 1 unknown.That will be of the form pxn = q so that xn = q/p.Substitute this value into one of the equations at the 2-equations-in-2-unknowns stage. That will give you xn-1.Work your way back to the top.The two methods are equivalent. There are shortcuts available for matrix inversion (eg using determinants), but these are too complicated to go into here.
unique number: The number 1 has only one factor. (It is therefore unique.)
The answer is 1.Here is the theorem:There is a unique circle passing through points P1 , P2 , P3 if and only if these three points are non-collinear.The proof is not too hard, but involves some linear algebra. I will post a link to it.
It is used for solving a system of linear equations where the number of equations equals the number of variables - and it is known that there is a unique solution.
Presumably the question concerned a PAIR of linear equations! The answer is two straight lines intersecting at the point whose coordinates are the unique solution.
This is the case when there is only one set of values for each of the variables that satisfies the system of linear equations. It requires the matrix of coefficients. A to be invertible. If the system of equations is y = Ax then the unique solution is x = A-1y.
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.
Cramer's rule is applied to obtain the solution when a system of n linear equations in n variables has a unique solution.
The equations are consistent and dependent with infinite solution if and only if a1 / a2 = b1 / b2 = c1 / c2.
So, take the case of two parallel lines, there is no solution at all. Now look at two equations that represent the same line, they have an infinite number of solutions. The solution is unique if and only if there is a single point of intersection. That point is the solution.
False, think of each linear equation as the graph of the line. Then the unique solution (one solution) would be the intersection of the two lines.
A single equation is several unknowns will rarely have a unique solution. A system of n equations in n unknown variables may have a unique solution.
A solution to an linear equation cx + d = f is in the form x = a for some a, we call a the solution (a might not be unique). Rewrite your sentence: x = 8, 8 is unique. So how many solution does it have?
There are three kinds:the equations have a unique solutionthe equations have no solutionthe equations have infinitely many solutions.