12 multiple-choice questions, progressively harder.
Solve the system x+y=5x + y = 5x+y=5 and x−y=1x - y = 1x−y=1.
Solution
Correct answer: B
Add the equations to eliminate yyy: (x+y)+(x−y)=5+1(x + y) + (x - y) = 5 + 1(x+y)+(x−y)=5+1.
2x=6⇒x=3,y=5−3=22x = 6 \Rightarrow x = 3,\quad y = 5 - 3 = 22x=6⇒x=3,y=5−3=2
The unique solution is (3,2)(3, 2)(3,2).
How many solutions does the system 2x−y=42x - y = 42x−y=4 and −4x+2y=3-4x + 2y = 3−4x+2y=3 have?
Correct answer: D
Put each in slope-intercept form: y=2x−4y = 2x - 4y=2x−4 and y=2x+32y = 2x + \tfrac{3}{2}y=2x+23. Same slope, different intercept.
m1=m2=2, b1≠b2⇒parallelm_1 = m_2 = 2,\ b_1 \ne b_2 \Rightarrow \text{parallel}m1=m2=2, b1=b2⇒parallel
So there is no solution.
Solve the system y=2x−1y = 2x - 1y=2x−1 and y=−x+5y = -x + 5y=−x+5.
Correct answer: C
Set the two expressions for yyy equal: 2x−1=−x+52x - 1 = -x + 52x−1=−x+5.
3x=6⇒x=2,y=2(2)−1=33x = 6 \Rightarrow x = 2,\quad y = 2(2) - 1 = 33x=6⇒x=2,y=2(2)−1=3
The solution is (2,3)(2, 3)(2,3).
The dependent system whose shared line is 2x−y=32x - y = 32x−y=3 has solutions of what form?
Solve the shared line 2x−y=32x - y = 32x−y=3 for yyy: y=2x−3y = 2x - 3y=2x−3. Any xxx is allowed, and yyy follows.
y=2x−3⇒(x, 2x−3)y = 2x - 3 \Rightarrow (x,\ 2x - 3)y=2x−3⇒(x, 2x−3)
Every such pair is a solution.
Solve the system x+2y=11x + 2y = 11x+2y=11 and x=y+2x = y + 2x=y+2.
Substitute x=y+2x = y + 2x=y+2 into x+2y=11x + 2y = 11x+2y=11: (y+2)+2y=11(y + 2) + 2y = 11(y+2)+2y=11.
3y=9⇒y=3,x=3+2=53y = 9 \Rightarrow y = 3,\quad x = 3 + 2 = 53y=9⇒y=3,x=3+2=5
The solution is (5,3)(5, 3)(5,3).
A system's elimination ends in 0=00 = 00=0. Which classification fits?
Correct answer: A
An always-true ending means the equations are the same line: solutions exist (consistent) and there are infinitely many (dependent).
0=0⇒consistent and dependent0 = 0 \Rightarrow \text{consistent and dependent}0=0⇒consistent and dependent
How many solutions does the system y=12x+4y = \tfrac{1}{2}x + 4y=21x+4 and 2y=x+82y = x + 82y=x+8 have?
Divide the second equation by 222: y=12x+4y = \tfrac{1}{2}x + 4y=21x+4, identical to the first.
2y=x+8 → ÷2 y=12x+42y = x + 8 \ \xrightarrow{\ \div 2\ }\ y = \tfrac{1}{2}x + 42y=x+8 ÷2 y=21x+4
Same line, so infinitely many solutions.
Which describes the solution set of x−y=2x - y = 2x−y=2 paired with 2x−2y=42x - 2y = 42x−2y=4?
The second equation is twice the first, so both are the line x−y=2x - y = 2x−y=2. Solve for yyy: y=x−2y = x - 2y=x−2.
(x, x−2) for every real x(x,\ x - 2)\ \text{for every real } x(x, x−2) for every real x
Every point on the line is a solution.
Which system is consistent and independent?
Consistent and independent means one solution, which requires different slopes.
y=x+1, y=−x+5: slopes 1 and −1y = x + 1,\ y = -x + 5:\ \text{slopes } 1 \text{ and } -1y=x+1, y=−x+5: slopes 1 and −1
Different slopes give a single crossing point. The other pairs are parallel or the same line.
Solve the system x+y=7x + y = 7x+y=7 and 2x−y=22x - y = 22x−y=2.
Add the equations to eliminate yyy: (x+y)+(2x−y)=7+2(x + y) + (2x - y) = 7 + 2(x+y)+(2x−y)=7+2.
3x=9⇒x=3,y=7−3=43x = 9 \Rightarrow x = 3,\quad y = 7 - 3 = 43x=9⇒x=3,y=7−3=4
The solution is (3,4)(3, 4)(3,4).
Solve the system 3x+y=103x + y = 103x+y=10 and y=x−2y = x - 2y=x−2.
Substitute y=x−2y = x - 2y=x−2 into 3x+y=103x + y = 103x+y=10: 3x+(x−2)=103x + (x - 2) = 103x+(x−2)=10.
4x=12⇒x=3,y=3−2=14x = 12 \Rightarrow x = 3,\quad y = 3 - 2 = 14x=12⇒x=3,y=3−2=1
The solution is (3,1)(3, 1)(3,1).
A consistent system is one that:
Consistent refers only to whether a solution exists, not how many. It means at least one solution, which covers both the one-solution and infinitely-many cases.
consistent⇔at least one solution\text{consistent} \Leftrightarrow \text{at least one solution}consistent⇔at least one solution
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