Explanation
Correct answer: C.
Choice C is correct. A system of two linear equations in two variables, and , has no solution if the graphs of the equations in the xy-plane are two distinct and parallel lines, which don't intersect. Two lines in the xy-plane are distinct and parallel if their slopes are the same and the y-coordinates of their y-intercepts are different. A linear equation can be written in slope-intercept form, , where is the slope of the line in the xy-plane and is the y-coordinate of the y-intercept. Choice C can be written in slope-intercept form by adding to both sides of the equation, which gives . In the equations and , the values of are each , and the values of are and , respectively. Since the slopes of these lines are the same and the y-coordinates of the y-intercepts are different, it follows that choice C could be the second equation in the system.
Desmos solution:
Note: Equal slopes and different intercepts make the line in choice C parallel to the blue line, so they never intersect.
Enter in Desmos:
Reviewed and rewritten by CookSAT
Why the other choices are wrong
Choice A
Choice A is incorrect. Adding to both sides of this equation gives . In the equations and , the values of aren't equal, so the slopes of these lines are different. Therefore, these two lines intersect and the corresponding system of equations has one solution.
Choice B
Choice B is incorrect. Adding to both sides of this equation gives . In the equations and , the values of aren't equal, so the slopes of these lines are different. Therefore, these two lines intersect and the corresponding system of equations has one solution.
Choice D
Choice D is incorrect. Adding to both sides of this equation gives . Since this equation is the same as the given equation, these two lines are not distinct. Therefore, the corresponding system of equations has infinitely many solutions.