1. Introduction
Two equations that are solved together are called systems of equations. The solution to a system of equations is the point or points that make both equations true. Systems of equations can have one solution, no solutions, or an infinite number of solutions. Finding the solution to a system of equations is important to many real-world applications.
2.2.1: Proving Equivalencies

2. Key Concepts
There are various methods to solving a system of equations. Two methods include the substitution method and the elimination method.
2.2.1: Proving Equivalencies

3. Key Concepts, continued
Solving Systems of Equations by Substitution
This method involves solving one of the equations for one of the variables and substituting that into the other equation.
2.2.1: Proving Equivalencies

4. Key Concepts, continued
Substitution Method
Solve one of the equations for one of the variables in terms of the other variable.
Substitute, or replace the resulting expression into the other equation.
Solve the equation for the second variable.
Substitute the found value into either of the original equations to find the value of the other variable.
2.2.1: Proving Equivalencies

5. Key Concepts, continued
Solutions to systems are written as an ordered pair, (x, y). This is where the lines would cross if graphed.
If the resulting solution is a true statement, such as 9 = 9, then the system has an infinite number of solutions. This is where lines would coincide if graphed.
2.2.1: Proving Equivalencies

6. Key Concepts, continued
If the result is an untrue statement, such as 4 = 9, then the system has no solutions. This is where lines would be parallel if graphed.
Check your answer by substituting the x and y values back into the original equations. If the answer is correct, the equations will result in true statements.
2.2.1: Proving Equivalencies

7. Key Concepts, continued
Solving Systems of Equations by Elimination Using Addition or Subtraction
This method involves adding or subtracting the equations in the system so that one of the variables is eliminated.
Properties of equality allow us to combine the equations by adding or subtracting the equations to eliminate one of the variables.
2.2.1: Proving Equivalencies

8. Key Concepts, continued
Elimination Method Using Addition or Subtraction
Add the two equations if the coefficients of one of the variables are opposites of each other.
Subtract the two equations if the coefficients of one of the variables are the same.
Solve the equation for the second variable.
Substitute the found value into either of the original equations to find the value of the other variable.
2.2.1: Proving Equivalencies

9. Key Concepts, continued
Solving Systems of Equations by Elimination Using Multiplication
This method is used when one set of variables are neither opposites nor the same. Applying the multiplication property of equality changes one or both equations.
Solving a system of equations algebraically will always result in an exact answer.
2.2.1: Proving Equivalencies

10. Key Concepts, continued
Elimination Method Using Multiplication
Multiply each term of the equation by the same number. It may be necessary to multiply the second equation by a different number in order to have one set of variables that are opposites or the same.
Add or subtract the two equations to eliminate one of the variables.
Solve the equation for the second variable.
Substitute the found value into either of the original equations to find the value of the other variable.
2.2.1: Proving Equivalencies

11. Common Errors/Misconceptions
finding the value for only one of the variables of the system
forgetting to distribute negative signs when substituting expressions for variables
forgetting to multiply each term by the same number when solving by elimination using multiplication
2.2.1: Proving Equivalencies

12. Guided Practice Example 2 Solve the following system by elimination.
2.2.1: Proving Equivalencies

13. Guided Practice: Example 2, continued
Add the two equations if the coefficients of one of the variables are opposites of each other.
3y and –3y are opposites, so the equations can be added. Add downward, combining like terms only.
2.2.1: Proving Equivalencies

14. Guided Practice: Example 2, continued Simplify. 3x = 0
2.2.1: Proving Equivalencies

15. Guided Practice: Example 2, continued
Solve the equation for the second variable.
3x = 0
x = 0
Divide both sides by 3.
2.2.1: Proving Equivalencies

16. Guided Practice: Example 2, continued
Substitute the found value, x = 0, into either of the original equations to find the value of the other variable.
2x – 3y = –11
First equation of the system
2(0) – 3y = –11
Substitute 0 for x.
– 3y = –11
Simplify.
Divide both sides by –3.
2.2.1: Proving Equivalencies

17. ✔ Guided Practice: Example 2, continued
The solution to the system of equations is
If graphed, the lines would cross at ✔
2.2.1: Proving Equivalencies

18. Guided Practice: Example 2, continued
2.2.1: Proving Equivalencies

19. Guided Practice Example 3
Solve the following system by multiplication.
2.2.1: Proving Equivalencies

20. Guided Practice: Example 3, continued
Multiply each term of the equation by the same number.
The variable x has a coefficient of 1 in the first equation and a coefficient of –2 in the second equation.
Multiply the first equation by 2.
x – 3y = 5
Original equation
2(x – 3y = 5)
Multiply the equation by 2.
2x – 6y = 10
2.2.1: Proving Equivalencies

21. Guided Practice: Example 3, continued
Add or subtract the two equations to eliminate one of the variables.
2.2.1: Proving Equivalencies

22. Guided Practice: Example 3, continued Simplify.
0 + 0 = 14
0 = 14
This is NOT a true statement.
2.2.1: Proving Equivalencies

23. ✔ Guided Practice: Example 3, continued
The system does not have a solution. There are no points that will make both equations true. ✔
2.2.1: Proving Equivalencies

24. Guided Practice: Example 3, continued
2.2.1: Proving Equivalencies