1.) Can a system of linear equations have exactly two solutions?
A. True
B. False
2.) The solution of a system of equations with no solution describes what kind of lines?
a.) parallel
b.) perpendicular
c.) skew
d.) coincidental
1. No
2. Parallel
1.) The answer is B. False.
To determine whether a system of linear equations can have exactly two solutions, we need to consider the nature of the equations.
A system of linear equations can have one of three possibilities: (1) no solution, (2) a unique solution, or (3) infinitely many solutions. It cannot have exactly two solutions.
If a system of linear equations has no solution, it means that the two lines represented by the equations are parallel and never intersect. In this case, the lines are "skew" (option c).
If a system of linear equations has a unique solution, it means that the two lines represented by the equations intersect at a single point. In this case, the lines are not parallel, perpendicular, or coincidental.
If a system of linear equations has infinitely many solutions, it means that the two lines represented by the equations are coincidental and overlap. In this case, the lines are coincidental (option d).
Therefore, the correct answer is B. False.
2.) The answer is a.) parallel.
A system of equations with no solution describes parallel lines.
When two lines are parallel, they have the same slope but different y-intercepts. If we write the equations of such lines in slope-intercept form (y = mx + b), the slopes (m) will be equal, while the y-intercepts (b) will be different.
Since parallel lines never intersect, there is no point of intersection, and hence, there is no solution to the system of equations.
Therefore, the correct answer is a.) parallel.