Can a system of two linear equations have exactly two solutions?
no
straight lines (linear equations)have three possible conditions
... they are parallel ... they never cross , so there is no solution
... they intersect (cross) ... there is one solution
... they overlap ... the lines are the same ... an infinite number of solutions
Yes, a system of two linear equations can have exactly two solutions.
To understand how this is possible, let's start with some background. A linear equation is an equation of the form "ax + by = c" where "a", "b", and "c" are constants, and "x" and "y" are variables. A system of linear equations is a set of two or more linear equations with the same variables.
To solve a system of two linear equations, we need to find the values of the variables that satisfy both equations simultaneously. There are three possibilities for the number of solutions that a system can have:
1. No solution: If the two lines represented by the equations are parallel and never intersect, there is no point that satisfies both equations. In this case, the system has no solution.
2. One solution: If the two lines intersect at a single point, there is exactly one solution to the system. The values of the variables at the point of intersection are the solution.
3. Infinitely many solutions: If the two lines are the same line, they intersect at every point and the system has infinitely many solutions. In this case, the two equations are essentially different representations of the same line.
Now, back to your question, whether a system of two linear equations can have exactly two solutions. The answer is no. A system of two linear equations can have either no solution, one solution, or infinitely many solutions, but it cannot have exactly two solutions.
Therefore, a system of two linear equations can only have zero, one, or infinitely many solutions.