12 multiple-choice questions, progressively harder.
In a system of two linear equations, the two lines cross at exactly one point. How many solutions does the system have?
Solution
Correct answer: C
A solution is a point lying on both lines, so the solutions are the points the two lines share. Lines that cross at one point share exactly that one point.
one shared point⇒one solution\text{one shared point} \Rightarrow \text{one solution}one shared point⇒one solution
So the system has exactly one solution.
The two lines of a system are parallel and never meet. How many solutions does the system have?
Correct answer: A
The solutions of a system are the points its lines share, and parallel lines share no point.
no shared point⇒no solution\text{no shared point} \Rightarrow \text{no solution}no shared point⇒no solution
So the system has no solution.
Which of the following is impossible for a linear system in two variables?
Correct answer: D
Two distinct lines meet in at most one point, and identical lines share all their points, so the only possible counts are none, one, or infinitely many.
possible: 0, 1, ∞\text{possible: } 0,\ 1,\ \inftypossible: 0, 1, ∞
Exactly two solutions can never occur.
While solving a system by elimination, the work reduces to x=7x = 7x=7. How many solutions does the system have?
The elimination ended at a single value of xxx, which forces one matching value of yyy by back-substitution.
x=7⇒one solutionx = 7 \Rightarrow \text{one solution}x=7⇒one solution
A system of equations that has at least one solution is called:
Correct answer: B
Consistency is about whether any solution exists at all. Having at least one solution is exactly what consistent means.
at least one solution⇒consistent\text{at least one solution} \Rightarrow \text{consistent}at least one solution⇒consistent
A system of equations that has no solution is called:
A system with no solution is inconsistent; the word consistent is reserved for systems that do have a solution.
no solution⇒inconsistent\text{no solution} \Rightarrow \text{inconsistent}no solution⇒inconsistent
A consistent system whose two equations describe the very same line is called:
When one equation is just a rescaling of the other, they carry the same information and are dependent.
same line⇒dependent (infinitely many)\text{same line} \Rightarrow \text{dependent (infinitely many)}same line⇒dependent (infinitely many)
How many solutions does the system y=2x+1y = 2x + 1y=2x+1 and y=3x−4y = 3x - 4y=3x−4 have?
The slopes are 222 and 333. Different slopes mean the lines cross once.
2≠3⇒one solution2 \ne 3 \Rightarrow \text{one solution}2=3⇒one solution
How many solutions does the system y=−x+3y = -x + 3y=−x+3 and y=−x+3y = -x + 3y=−x+3 have?
The two equations are identical, so they graph as the same line.
same line⇒infinitely many solutions\text{same line} \Rightarrow \text{infinitely many solutions}same line⇒infinitely many solutions
Is (2,1)(2, 1)(2,1) a solution of the system x+y=3x + y = 3x+y=3 and x−y=1x - y = 1x−y=1?
A solution of a system must satisfy both equations, so test the pair in each.
2+1=3 ✓2−1=1 ✓2 + 1 = 3 \ \checkmark \qquad 2 - 1 = 1 \ \checkmark2+1=3 ✓2−1=1 ✓
Both hold, so (2,1)(2, 1)(2,1) solves the system.
A system is described as inconsistent. Which statement must be true?
Inconsistent means no ordered pair satisfies both equations.
inconsistent⇒no solution\text{inconsistent} \Rightarrow \text{no solution}inconsistent⇒no solution
A dependent system of two linear equations has how many solutions?
A dependent system is two names for the same line, so every point on that line is a solution.
dependent⇒infinitely many solutions\text{dependent} \Rightarrow \text{infinitely many solutions}dependent⇒infinitely many solutions
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