A system of linear equations asks for points shared by two lines. Exactly one solution means one intersection, so compare the lines’ slopes: different slopes guarantee one crossing. Graph the candidate pair in Desmos to see it. Matching slopes give either parallel lines or the same line; one visible line can hide infinitely many solutions.
Hints
- Hint 1
A solution is a point that makes both equations true. On a graph, that point lies where the lines meet. What happens if the lines never meet, or if they lie on top of each other?
- Hint 2
The slope is the number multiplying in an equation written as . Lines with different slopes cross once. Compare slopes first, then watch for equations that describe the same line.
Step-by-step
Compare the lines, then graph the crossing
Step 1Rule out parallel slanted lines
A slope tells how much changes when increases by . In , the slope is . The lines and have the same slope, , but meet the -axis at different heights. They stay parallel and never cross.
- Step 2
Rule out separate vertical lines
The equation draws a vertical line whose points all have ; draws another. No point can have both -values, so these lines never meet.
- Step 3
Spot two equations for the same line
Double every term of :
That gives the other equation in the pair. The two equations draw the same line, so every point on it is a solution, not just one point.
- Step 4
Confirm the one crossing in Desmos
The remaining lines have slopes and , so they cross exactly once. Type and on separate lines in Desmos. Click their crossing; Desmos labels it about . That shared point makes both equations true. Different slopes guarantee exactly one solution. The system is and . Choice B.