#1 · Graph it, then zoom out
The question. How many solutions does this system of equations have?
The shortcut. A solution to a system is a point where the two graphs meet: an intersection point. (A point where they touch without crossing still counts as one.) So instead of solving, graph both equations in the calculator built into Bluebook and count the intersection points. It's often the fastest way to answer a "how many solutions" question.
The fix
Don't count only the intersections in the first window. Zoom out and check farther along both graphs. If the view is still ambiguous, set the equations equal:
That confirms intersections at (0, 0) and (10, 20): two solutions.
Two related cases
Two question types from our own bank that look like this one, and how the shortcut changes for each:
- An unknown constant. "For what value of k does the system have exactly one solution?" There's nothing concrete to graph until you know k. You can add a slider and drag it until the graphs just touch (that's shortcut #11), or set the discriminant equal to zero.
- No solution or infinitely many solutions for two lines. When every number is known, graphing works here too: lines that look parallel mean no solution, and if you entered two equations but see only one line, they're the same line (infinitely many solutions; hide one to check). But "looks parallel" is this episode's trap again: lines with slopes 2 and 2.01 look parallel up close and still meet far off the screen, so zoom out or compare the slopes before you answer. When the question has an unknown constant, compare slopes and intercepts, or, in ax + by = c form, check whether the coefficients are in the same ratio.
The takeaway: if you graph a system, check beyond the first window before counting intersections.