Solving Systems of Equations by Substitution

Solving Systems of Equations by Substitution

Overview

• The process of solving systems of equations can be done using multiple methods, one of which is substitution.

What is substitution?

• Substitution is a method used to solve a system of equations by solving one equation for one variable and then substituting that expression into another equation.

Steps to Solve Systems by Substitution

1. Choose one of the equations and solve it for one variable in terms of the other.

2. Substitute the expression obtained from step 1 into the other equation. This will yield an equation with only one variable.

3. Solve the new equation for the remaining variable.

4. Substitute back to find the other variable.

5. Check the solution by substituting both values back into the original equations.

Worksheet Problems

System of equations examples to solve using substitution:

1. Problem 1:

   • Equation 1: y = 6x - 11

   • Equation 2: -2x - 3y = -7

   • Expected solution: (2, 1)

2. Problem 2:

   • Equation 1: 2x - 3y = -1

   • Equation 2: y = x - 1

   • Expected solution: (4, 3)

3. Problem 3:

   • Equation 1: y = -3x + 5

   • Equation 2: 5x - 4y = -3

   • Expected solution: (1, 2)

4. Problem 4:

   • Equation 1: -3x - 3y = 3

   • Equation 2: y = -5x - 17

   • Expected solution: (-4, 3)

5. Problem 5:

   • Equation 1: y = -2

   • Equation 2: 4x - 3y = 18

   • Expected solution: (3, -2)

6. Problem 6:

   • Equation 1: y = 5x - 7

   • Equation 2: -3x - 2y = -12

   • Expected solution: (2, 3)

7. Problem 7:

   • Equation 1: -4x + y = 6

   • Equation 2: -5x - y = 21

   • Expected solution: (-3, -6)

8. Problem 8:

   • Equation 1: -7x - 2y = -13

   • Equation 2: x - 2y = 11

   • Expected solution: (3, -4)

9. Problem 9:

   • Equation 1: -5x + y = -2

   • Equation 2: -3x + 6y = -12

   • Expected solution: (0, -2)

10. Problem 10:

    • Equation 1: -5x + y = -3

    • Equation 2: 3x - 8y = 24

    • Expected solution: (0, -3)

Additional Problems

More systems to be solved:

11. Equation 1: x + 3y = 1
    Equation 2: -3x - 3y = -15

    • Expected solution: (7, -2)

12. Equation 1: -3x - 8y = 20
    Equation 2: -5x + y = 19

    • Expected solution: (-4, -1)

13. Equation 1: -3x + 3y = 4
    Equation 2: -x + y = 3

    • Expected solution: No solution

14. Equation 1: -3x + 3y = 3
    Equation 2: -5x + y = 13

    • Expected solution: (-3, -2)

15. Equation 1: 6x + 6y = -6
    Equation 2: 5x + y = -13

    • Expected solution: (-3, 2)

16. Equation 1: 2x + y = 20
    Equation 2: 6x - 5y = 12

    • Expected solution: (7, 6)

17. Equation 1: -3x - 4y = 2
    Equation 2: 3x + 3y = -3

    • Expected solution: (-2, 1)

18. Equation 1: -2x + 6y = 6
    Equation 2: -7x + 8y = -5

    • Expected solution: (3, 2)

19. Equation 1: -5x - 8y = 17
    Equation 2: 2x - 7y = -17

    • Expected solution: (-5, 1)

20. Equation 1: -2x - y = -9
    Equation 2: 5x - 2y = 18

    • Expected solution: (4, 1)

Conclusion

• The substitution method is a valuable technique in algebra for solving systems of linear equations, allowing one to isolate variables and systematically arrive at solutions. By practicing these problems, one can become proficient at employing this method effectively.