Systems of Equations - ACT Math

1. Solve the system of linear equations: $\begin{cases} 2x + y = 11 \\ x - y = 1 \end{cases}$ What is the solution $ (x, y) $?
   Answer: $(4,3)$ Use elimination method: add the equations $2x + y = 11$ and $x - y = 1$. This gives $3x = 12$, so $x = 4$. Substitute $x = 4$ into $x - y = 1$: $4 - y = 1$, so $y = 3$. The solution is $(4, 3)$. Choice B reverses the x and y values.
2. A taxi charges a flat fee plus a per-mile rate. A 4-mile ride costs $14 and a 7-mile ride costs $20. If $x$ is the flat fee and $y$ is the cost per mile, what is $(x,y)$?
   Answer: $(6,2)$ Set up the system where $x$ is flat fee and $y$ is per-mile rate: $x + 4y = 14$ and $x + 7y = 20$. Subtract the first from the second: $3y = 6$, so $y = 2$. Substitute back: $x + 4(2) = 14$, so $x = 6$. The solution is $(6, 2)$.
3. What is the solution $(x, y)$ to the system $x + y = 6$ and $2x - y = 3$?
   Answer: (3, 3) Use elimination to solve this system. Adding the equations x + y = 6 and 2x - y = 3 eliminates y: 3x = 9, so x = 3. Substituting x = 3 into the first equation: 3 + y = 6, so y = 3. The solution is (3, 3).
4. If the system below is true, what is the value of $y$? $\begin{cases} 3x + 4y = 1\\ 6x - 4y = 11 \end{cases}$
   Answer: $-\tfrac{3}{4}$ Use elimination by adding the equations. Adding 3x + 4y = 1 and 6x - 4y = 11 gives 9x = 12, so x = 4/3. Substitute into 3x + 4y = 1: 3(4/3) + 4y = 1, which gives 4 + 4y = 1, so 4y = -3 and y = -3/4.
5. If $2x + 3y = 13$ and $2x - 3y = 1$, what is the value of $y$?
   Answer: 2 Use the elimination method by adding $2x + 3y = 13$ and $2x - 3y = 1$. This eliminates y, giving $4x = 14$, so $x = 14/4 = 3.5$. Substitute into $2x - 3y = 1$: $2(3.5) - 3y = 1$, $7 - 3y = 1$, $-3y = -6$, $y = 2$. Alternatively, subtract the equations to get $6y = 12$, $y = 2$ directly. Choice C of 3 might come from misadding to $4x = 12$.
6. Solve the system: $\begin{cases}2x+y=11\\x-y=1\end{cases}$ What is the solution $(x, y)$?
   Answer: $(4,3)$ Use elimination to solve this system by adding the equations. From $2x + y = 11$ and $x - y = 1$, adding gives $3x = 12$, so $x = 4$. Substitute back: $4 - y = 1$, so $y = 3$. The solution is $(4, 3)$.
7. Solve the system: $\begin{cases} x + y = 8\\ 3x - y = 4 \end{cases}$ What is the solution $(x, y)$?
   Answer: $(3, 5)$ Use elimination: Add the equations $x + y = 8$ and $3x - y = 4$. This gives $4x = 12$, so $x = 3$. Substitute $x = 3$ into $x + y = 8$: $3 + y = 8$, so $y = 5$. The solution is $(3, 5)$.
8. Two numbers have sum 13 and difference 5. If $x+y=13$ and $x-y=5$, what is $(x,y)$?
   Answer: $(9,4)$ Use elimination to solve this system with $x + y = 13$ and $x - y = 5$. Adding the equations eliminates $y$: $2x = 18$, so $x = 9$. Substitute back: $9 + y = 13$, so $y = 4$. The solution is $(9, 4)$.
9. At a school store, 2 notebooks and 3 pens cost $12, and 1 notebook and 2 pens cost $7. If $x$ is the cost of a notebook and $y$ is the cost of a pen, what is $(x,y)$?
   Answer: $(3,2)$ Set up the system: $2x + 3y = 12$ and $x + 2y = 7$. Use elimination by multiplying the second equation by 2: $2x + 4y = 14$. Subtract the first equation: $y = 2$. Substitute back: $x + 2(2) = 7$, so $x = 3$. The solution is $(3, 2)$.
10. Solve the system: $\begin{cases} 4x + y = 9 \\ 2x - y = 3 \end{cases}$ Which ordered pair satisfies both equations?
    Answer: $(2, 1)$ Use elimination: Add the equations $4x + y = 9$ and $2x - y = 3$. This gives $6x = 12$, so $x = 2$. Substitute $x = 2$ into $2x - y = 3$: $4 - y = 3$, so $y = 1$. The solution is $(2, 1)$.
11. If $x + y = 8$ and $x - y = 4$, what is the value of $x$?
    Answer: 6 Use elimination to solve this system. Adding $x + y = 8$ and $x - y = 4$ eliminates y: $2x = 12$, so $x = 6$. Substituting into $x + y = 8$: $6 + y = 8$, so $y = 2$. Therefore, $x = 6$.
12. Which ordered pair satisfies both equations $y = x - 2$ and $x + y = 8$?
    Answer: (5, 3) Use substitution to solve this system. Substitute y = x - 2 into x + y = 8: x + (x - 2) = 8. This simplifies to 2x - 2 = 8, so x = 5. Then y = 5 - 2 = 3. The solution is (5, 3).
13. Solve the system: $\begin{cases} 2x + y = 11\\ x - y = 1 \end{cases}$ What is the solution $(x, y)$?
    Answer: $(4, 3)$ Use the elimination method by adding the equations 2x + y = 11 and x - y = 1. This eliminates y, giving 3x = 12, so x = 4. Substitute x = 4 into x - y = 1 to get 4 - y = 1, so y = 3. The solution is (4, 3). Choice B might arise from subtracting the equations instead, leading to x = 10/1 = 10, which is incorrect.
14. If $4x + y = 10$ and $2x - y = 2$, what is the value of $x$?
    Answer: 2 Use elimination to solve this system. Add the two equations: (4x + y) + (2x - y) = 10 + 2, which gives 6x = 12, so x = 2. This matches choice B.
15. Solve the system of equations: $\begin{cases} 2x + y = 11\\ x - y = 1 \end{cases}$ What is the solution $(x, y)$?
    Answer: $(4, 3)$ Use elimination method by adding the equations. Adding 2x + y = 11 and x - y = 1 gives 3x = 12, so x = 4. Substitute x = 4 into x - y = 1: 4 - y = 1, so y = 3. The solution is (4, 3). Choice B reverses the x and y values.
16. Solve the system: $\begin{cases} x + 2y = 10\\ 3x - 2y = 6 \end{cases}$ Which ordered pair satisfies both equations?
    Answer: $ (4,3) $ Use elimination: add equations $x + 2y = 10$ and $3x - 2y = 6$. The y terms cancel, giving $4x = 16$, so $x = 4$. Substitute $x = 4$ into $x + 2y = 10$: $4 + 2y = 10$, so $y = 3$. The solution is $ (4, 3) $.
17. Solve the system of equations: $\begin{cases} 5x - y = 14\\ 2x + y = 7 \end{cases}$ What is the solution $(x, y)$?
    Answer: $(3, 1)$ Use elimination by adding the equations directly. Adding 5x - y = 14 and 2x + y = 7 gives 7x = 21, so x = 3. Substitute x = 3 into 2x + y = 7: 2(3) + y = 7, so y = 1. The solution is (3, 1).
18. Solve the system: $3x + 2y = 12$ and $x - y = 1$.
    Answer: (3, 2) Use substitution to solve this system. From x - y = 1, we get x = y + 1. Substituting into 3x + 2y = 12: 3(y + 1) + 2y = 12, which simplifies to 5y + 3 = 12, so y = 2. Then x = 2 + 1 = 3. The solution is (3, 2).
19. What is the solution $(x, y)$ to the system $x + y = 9$ and $x - y = 1$?
    Answer: (5, 4) Use elimination to solve this system. Adding x + y = 9 and x - y = 1 eliminates y: 2x = 10, so x = 5. Substituting into x + y = 9: 5 + y = 9, so y = 4. The solution is (5, 4).
20. What is the solution $(x, y)$ to the system $x + y = 10$ and $x - y = 2$?
    Answer: (6, 4) Use elimination to solve this system. Add the two equations: (x + y) + (x - y) = 10 + 2, which gives 2x = 12, so x = 6. Substitute back into the first equation: 6 + y = 10, so y = 4. The solution is (6, 4), which matches choice A.

Start this quiz