Math 1 Quiz: Solving Systems By Elimination

What this quiz covers

This quiz focuses on Solving Systems By Elimination, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 1.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. The addition $3x + 4y + 2x - 4y = 20 + (-5)$ should yield $5x = 25$, not $5x = 15$
2. The substitution $3(3) + 4y = 20$ is correct, but solving $4y = 11$ should give $y = 2.75$
3. The check in the second equation should be $2(3) - 4(\frac{11}{4}) = 6 - 11 = -5$, which is incorrect
4. All steps are mathematically correct and the solution $(3, \frac{11}{4})$ satisfies both original equations (correct answer)

Question 2

1. The system has exactly one solution since subtracting eliminates $y$ and leaves a solvable equation in $x$ (correct answer)
2. The system has no solution because the $y$-coefficients are equal but the $x$-coefficients are different
3. The system has infinitely many solutions since both equations have the same $y$-coefficient $qty$
4. The conclusion depends on whether $tr = su$, which determines if the constant terms create a contradiction

Question 3

1. The system has no solution because the elimination process created a contradiction in the $x$-coefficients
2. The system has exactly one solution, and $y$ can be found by substituting $x = \frac{CE - BF}{AE - BD}$ back into either equation (correct answer)
3. The system has infinitely many solutions since $AE - BD \neq 0$ means the equations are scalar multiples
4. The conclusion depends on whether $CE - BF = 0$, which determines if the system is consistent or inconsistent

Question 4

1. When $ps - tn = 0$, because this creates the indeterminate form $0 = 0$ after elimination
2. When $ms + rn = 0$, because this makes the coefficient of $x$ zero after addition
3. When $ms = rn$, because the $x$-coefficients become equal and don't eliminate as intended (correct answer)
4. When $ns + sr = 0$, because the $y$-coefficients don't properly eliminate to zero

Question 5

1. Add the resulting equations $21x - 6y = 39$ and $21x + 35y = 28$ to get $42x + 29y = 67$
2. Subtract the first result from the second: $(21x + 35y) - (21x - 6y) = 28 - 39$ to eliminate $x$
3. Subtract the second result from the first: $(21x - 6y) - (21x + 35y) = 39 - 28$ to eliminate $x$
4. Recognize that both equations now have $21x$, so either subtraction order will eliminate $x$ successfully (correct answer)

Question 6

1. Multiply the first equation by $bcd$ and the second by $abd$, then eliminate the variable with simpler resulting coefficients
2. Cross-multiply each equation to eliminate fractions, then apply elimination to $bx + ay = ab$ and $dx + cy = cd$
3. Multiply the first equation by $d$ and the second by $-b$, then add to eliminate $y$ directly
4. Clear denominators by multiplying the first equation by $ab$ and the second by $cd$, creating integer coefficients before elimination (correct answer)

Question 7

1. $11x = 30$ (correct answer)
2. $11x = 26$
3. $x = 26$
4. $-x = -26$

Question 8

1. Only Student A will eliminate a variable completely on the first addition
2. Only Student B will eliminate a variable completely on the first addition
3. Both students will eliminate a variable completely on the first addition (correct answer)
4. Neither student will eliminate a variable completely on the first addition

Question 9

1. The conclusion is always correct when $0 = 0$ is obtained during elimination
2. The conclusion is correct only if the original equations are scalar multiples of each other (correct answer)
3. The conclusion is correct only if both original equations have the same slope when graphed
4. The conclusion is never correct; $0 = 0$ indicates a computational error was made

Question 10

1. $ms \neq nr$ and the system has infinitely many solutions
2. $ms \neq nr$ and the system has exactly one solution
3. $ms = nr$ and the system has no solutions
4. $ms = nr$ and the system has exactly one solution (correct answer)

Question 11

1. 2
2. 7 (correct answer)
3. 14
4. 5

Question 12

1. Multiply the first equation by $2$ and subtract it from the second equation to eliminate $x$
2. Multiply the second equation by $3$ and add it to the first equation to eliminate $y$ (correct answer)
3. Multiply the first equation by $-2$ and add it to the second equation to eliminate $x$
4. Multiply the second equation by $2$ and subtract the first equation to eliminate $x$

Question 13

1. Multiply the first equation by $6$ and the second by $12$ to clear all denominators completely
2. Multiply the first equation by $2$ and add it to the second equation to eliminate the $y$-terms
3. Clear denominators in each equation separately, then apply standard elimination techniques to the resulting system (correct answer)
4. Multiply the second equation by $2$ and add it to the first equation to eliminate the $y$-terms

Question 14

1. $b = 2$, making the left sides of the equations proportional while the right sides are not (correct answer)
2. $b = 8$, creating coefficients that eliminate to give a contradiction like $0 = 3$
3. $b = \frac{3}{2}$, ensuring the coefficient ratios are equal but the constant ratio is different
4. $b$ can be any value except $2$, since $b = 2$ would create infinitely many solutions

Question 15

1. Multiply the first equation by 2 and leave the second equation unchanged
2. Multiply both equations by their respective least common denominators
3. Convert all fractions to decimals before proceeding with elimination
4. Multiply the first equation by 6 and the second equation by 12 (correct answer)

Question 16

1. $k = 1$, because this makes the first equation become $x + 3y = 9$, which is exactly half the second (correct answer)
2. $k = 2$, because this makes both equations have the same $x$-coefficient when properly aligned
3. $k = 6$, because this creates proportional coefficients that eliminate to give the identity $0 = 0$
4. $k = 3$, because this makes the ratio of $x$-coefficients equal the ratio of $y$-coefficients