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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.

Q: Does every pair of linear simultaneous equations have a unique solution?

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simultaneous equations

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.

When (the graph of the equations) the two lines intersect. The equations will tell you what the slopes of the lines are, just look at them. If they are different, then the equations have a unique solution..

By definition, there cannot be a simultaneous equation that cannot be solved, there must be a set of simultaneous equations. It is important to realise that simultaneous equations need not be linear.It is simple to devise a pair of linear equations that are inconsistent:x + y = 1 and x + y = 2There is no solution. Graphically, the two lines are parallel.Another possibility isx + y = 1 and 2x + 2y = 2In this case there are an infinite number of solutions. Graphically, the two lines are coincidet, so that every point on the common line is a solution. There is, therefore, no unique solution.Yet another situation can arise when the domain of the equations is restricted.For example,x2 + y2 = -1 where x and y are real along with any other equation in x and y.

It means that at least one of the equations can be expressed as a linear combination of some of the other equations. A linear combination of equations is the addition (or subtraction) of equations. And since an equation can be added several times, it includes multiples of equations. For example, if you have x + 2y = 3 and 2x + y = 4 Then adding 2 times the first and 3 times the second gives 8x + 7y = 18 This is, therefore, dependent on the other 2. If you have n unknown variables, there will be a unique solution if, and only if, you must have a set of n independent linear equations.

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simultaneous equations

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.

When (the graph of the equations) the two lines intersect. The equations will tell you what the slopes of the lines are, just look at them. If they are different, then the equations have a unique solution..

The equations are consistent and dependent with infinite solution if and only if a1 / a2 = b1 / b2 = c1 / c2.

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.

By definition, there cannot be a simultaneous equation that cannot be solved, there must be a set of simultaneous equations. It is important to realise that simultaneous equations need not be linear.It is simple to devise a pair of linear equations that are inconsistent:x + y = 1 and x + y = 2There is no solution. Graphically, the two lines are parallel.Another possibility isx + y = 1 and 2x + 2y = 2In this case there are an infinite number of solutions. Graphically, the two lines are coincidet, so that every point on the common line is a solution. There is, therefore, no unique solution.Yet another situation can arise when the domain of the equations is restricted.For example,x2 + y2 = -1 where x and y are real along with any other equation in x and y.

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.

It means that at least one of the equations can be expressed as a linear combination of some of the other equations. A linear combination of equations is the addition (or subtraction) of equations. And since an equation can be added several times, it includes multiples of equations. For example, if you have x + 2y = 3 and 2x + y = 4 Then adding 2 times the first and 3 times the second gives 8x + 7y = 18 This is, therefore, dependent on the other 2. If you have n unknown variables, there will be a unique solution if, and only if, you must have a set of n independent linear equations.

There are three kinds:the equations have a unique solutionthe equations have no solutionthe equations have infinitely many solutions.