Systems of Equations Worksheet

Question 1

Solve the system using any method: y = 2x + 1 and y = x + 4.

Accepted Answer

The solution is x = 3 and y = 7. Since both expressions equal y, set 2x + 1 equal to x + 4 and solve to get x = 3, then substitute to find y = 7.

Question 2

Solve the system: x + y = 10 and x - y = 2.

Accepted Answer

The solution is x = 6 and y = 4. Adding the equations gives 2x = 12, so x = 6, and then substituting into x + y = 10 gives y = 4.

Question 3

Solve the system: 2x + 3y = 12 and x - y = 1.

Accepted Answer

The solution is x = 3 and y = 2. From x - y = 1, write x = y + 1, then substitute into 2x + 3y = 12 to get 2(y + 1) + 3y = 12, which leads to y = 2 and x = 3.

Question 4

Solve the system: 3x + 2y = 16 and 5x - 2y = 8.

Accepted Answer

The solution is x = 3 and y = 3. Adding the equations eliminates y and gives 8x = 24, so x = 3, and substituting back gives y = 3.

Question 5

Solve the system: y = -x + 5 and 2x + y = 8.

Accepted Answer

The solution is x = 3 and y = 2. Substitute y = -x + 5 into 2x + y = 8 to get 2x + (-x + 5) = 8, so x = 3 and then y = 2.

Question 6

Solve the system: 4x - y = 11 and 2x + y = 1.

Accepted Answer

The solution is x = 2 and y = -3. Adding the equations gives 6x = 12, so x = 2, and substituting into 2x + y = 1 gives y = -3.

Question 7

Determine whether the system has one solution, no solution, or infinitely many solutions: y = 3x - 2 and y = 3x + 4.

Accepted Answer

The system has no solution. The lines have the same slope but different y-intercepts, so they are parallel and never intersect.

Question 8

Determine whether the system has one solution, no solution, or infinitely many solutions: 2x + 4y = 10 and x + 2y = 5.

Accepted Answer

The system has infinitely many solutions. The first equation is exactly twice the second equation, so both equations represent the same line.

Question 9

Solve the system: 3x + y = 7 and 2x - y = 3.

Accepted Answer

The solution is x = 2 and y = 1. Adding the equations gives 5x = 10, so x = 2, and substituting into 3x + y = 7 gives y = 1.

Question 10

Solve the system: x + 2y = 9 and 3x - 2y = 7.

Accepted Answer

The solution is x = 4 and y = 2. Adding the equations eliminates y and gives 4x = 16, so x = 4, and then substituting gives y = 2.

Question 11

A school sells tickets to a play. Student tickets cost 5 dollars and adult tickets cost 8 dollars. A total of 22 tickets were sold for 149 dollars. Write and solve a system to find how many student tickets and adult tickets were sold.

Accepted Answer

The solution is 9 student tickets and 13 adult tickets. Let s be student tickets and a be adult tickets. Then s + a = 22 and 5s + 8a = 149. Substituting s = 22 - a into the second equation gives 5(22 - a) + 8a = 149, so a = 13 and s = 9.

Question 12

The sum of two numbers is 18. Their difference is 4. Write and solve a system to find the numbers.

Accepted Answer

The numbers are 11 and 7. Let x and y be the numbers. Then x + y = 18 and x - y = 4. Adding the equations gives 2x = 22, so x = 11 and y = 7.

Question 13

Solve the system: y = 4x - 7 and y = -2x + 11.

Accepted Answer

The solution is x = 3 and y = 5. Set 4x - 7 equal to -2x + 11, solve to get x = 3, and substitute to find y = 5.

Question 14

Solve the system: 2x + 5y = 4 and 4x + 10y = 8.

Accepted Answer

The system has infinitely many solutions. The second equation is exactly twice the first equation, so both equations describe the same line.

Question 15

Solve the system: x - 3y = -11 and 2x + 3y = 7.

Accepted Answer

The solution is x = -4 and y = 1. Adding the equations gives 3x = -4, which would be incorrect if added directly because the constants are -11 and 7. Instead, add the equations carefully: (x - 3y) + (2x + 3y) = -11 + 7 gives 3x = -4, so x = -4/3. Substituting into x - 3y = -11 gives y = 29/9. Therefore, the correct solution is x = -4/3 and y = 29/9.

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