Systems of Equations
Solving pairs of linear equations
Systems of Equations
Solving pairs of linear equations
Math - Grade 9-12
- 1
Solve the system using any method: y = 2x + 1 and y = x + 4.
Set the two expressions for y equal to each other.
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. - 2
Solve the system: x + y = 10 and x - y = 2.
Try adding the two equations together.
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. - 3
Solve the system: 2x + 3y = 12 and x - y = 1.
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. - 4
Solve the system: 3x + 2y = 16 and 5x - 2y = 8.
The y-terms are opposites, so elimination will work quickly.
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. - 5
Solve the system: y = -x + 5 and 2x + y = 8.
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. - 6
Solve the system: 4x - y = 11 and 2x + y = 1.
Add the equations to eliminate y.
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. - 7
Determine whether the system has one solution, no solution, or infinitely many solutions: y = 3x - 2 and y = 3x + 4.
The system has no solution. The lines have the same slope but different y-intercepts, so they are parallel and never intersect. - 8
Determine whether the system has one solution, no solution, or infinitely many solutions: 2x + 4y = 10 and x + 2y = 5.
Check whether one equation is a multiple of the other.
The system has infinitely many solutions. The first equation is exactly twice the second equation, so both equations represent the same line. - 9
Solve the system: 3x + y = 7 and 2x - y = 3.
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. - 10
Solve the system: x + 2y = 9 and 3x - 2y = 7.
The y-coefficients are opposites.
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. - 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.
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. - 12
The sum of two numbers is 18. Their difference is 4. Write and solve a system to find the numbers.
Use one equation for the sum and one for the difference.
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. - 13
Solve the system: y = 4x - 7 and y = -2x + 11.
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. - 14
Solve the system: 2x + 5y = 4 and 4x + 10y = 8.
Compare the second equation to the first by multiplying.
The system has infinitely many solutions. The second equation is exactly twice the first equation, so both equations describe the same line. - 15
Solve the system: x - 3y = -11 and 2x + 3y = 7.
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.