Does the system of equations 3x + 4y = 8 and x + y = 2 have any solutions.
Each of the given equations 3x + 4y = 8 and x + y = 2 represents a straight line. The solution of the system of equations is the points at which the lines represented by the two equations intercept.
If the two equations represent parallel lines, there is no solution. If the two equations represent the same line there are an infinite number of solutions and in all other cases there is a single unique solution.
Writing 3x + 4y = 8 in the slope intercept form y = (-3/4)x + 2. The slope of the line is -3/4. The slope of the line represented by x + y = 2 is -1. As the equations represent two unique lines that are not parallel, there is a unique solution for the given system of equations.
The given system of equations has one unique solution.