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Q: What system of two linear equations in two variables that has exactly one solution?

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The solution of a system of linear equations is a pair of values that make both of the equations true.

A system of equations may have any amount of solutions. If the equations are linear, the system will have either no solution, one solution, or an infinite number of solutions. If the equations are linear AND there are as many equations as variables, AND they are independent, the system will have exactly one solution.

simultaneous equations

A system of linear equations can only have: no solution, one solution, or infinitely many solutions.

A linear equation in two variables will not have a single solution. Its solution set is a line in the Cartesian plane. The solution to non-linear equations will depend on the equation.

The solution to a system on linear equations in nunknown variables are ordered n-tuples such that their values satisfy each of the equations in the system. There need not be a solution or there can be more than one solutions.

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

The slopes (gradients) of the two equations are different.

perpendicular

Yes. The easiest case to see where this is true is in the case that the equations are all of degree = 1, which will yield one solution per variable.

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.

They are the simplest form of relationship between two variables. Non-linear equations are often converted - by transforming variables - to linear equations.

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