What type of lines would make up a system with no solution?
A) perpendicular lines
B) intersecting lines
C) the same line
D) parallel lines
D) parallel lines
Yeah, parallel
The correct answer is D) parallel lines.
To understand why parallel lines make up a system with no solution, let's start by understanding what a system of equations is. A system of equations consists of two or more equations that are solved simultaneously to find the values of the variables that satisfy all the equations.
When dealing with a system of linear equations, the graphs of the equations represent lines on a coordinate plane. These lines can be classified into three categories: perpendicular, intersecting, or parallel.
Perpendicular lines are lines that meet or intersect at a right angle, forming a 90-degree angle between them.
Intersecting lines are lines that cross or meet at a single point on the coordinate plane. In this case, the system of equations has a unique solution where the coordinates of the point represent the values of the variables that satisfy the equations.
The same line represents an equation or multiple equations that are equivalent or redundant. In this case, the system of equations has infinitely many solutions since any point on the line satisfies the equations.
Now, let's focus on parallel lines. Parallel lines are lines that never intersect, meaning they are always the same distance apart and have the same slope. In a system of linear equations, if the lines are parallel, it means they have the same slope but different y-intercepts. When trying to solve such a system, there will be no point of intersection, resulting in no solution. This occurs because parallel lines, by definition, do not intersect, and therefore there are no values for the variables that satisfy both equations simultaneously.
So, in summary, a system of equations with parallel lines has no solution, making option D) parallel lines the correct answer.