You are solving a system of two linear equations in two variables, and you discover that there are no solutions to the system. Which of the following graphs could describe the system of equations?
The Correct Answer and Explanation is:
The correct answer is D, the Pink graph.
A system of linear equations consists of two or more linear equations that are considered together. The solution to such a system is the set of values for the variables that makes all the equations true simultaneously. When we represent these linear equations graphically, each equation forms a straight line on a coordinate plane. The solution to the system is found at the point where these lines intersect.
There are three possible outcomes when graphing a system of two linear equations in two variables. First, the lines can intersect at a single point, which means there is exactly one unique solution. Second, the lines can be coincident, meaning they are the same line. In this case, there are infinitely many solutions, as every point on the line satisfies both equations. The third possibility, which is relevant to this question, is that the lines are parallel and distinct. Parallel lines have the same slope but different y-intercepts, so they never cross or touch each other. Because there is no point of intersection, a system of parallel lines has no solution.
The problem states that we have discovered a system with no solutions. Therefore, we must identify the graph that shows two parallel lines.
Let’s examine the options provided. The Blue graph, the Green graph, and the Purple graph each show two lines that clearly intersect at one point. These three graphs represent systems with one unique solution. In contrast, the Pink graph displays two distinct horizontal lines. Horizontal lines have a slope of zero. Since both lines have the same slope (zero) but different y-intercepts (one at y=3 and the other at y=1), they are parallel. Because they are parallel, they will never intersect. This graphical representation perfectly matches the description of a system of linear equations with no solution.
