1. Systems of Equations 7-4 Learn to solve systems of equations.

2. Systems of Equations 7-4 A system of equations is a set of two or more equations that contain two or more variables. A solution of a system of equations is a set of values that are solutions of all of the equations. If the system has two variables, the solutions can be written as ordered pairs.

3. Systems of Equations 7-4 When solving systems of equations, remember to find values for all of the variables. Caution!

4. Systems of Equations 7-4 Additional Example 1A: Solving Systems of Equations Solve the system of equations. y = 4x – 6 y = x + 3 y = 4x – 6y = x + 3 The expressions x + 3 and 4x – 6 both equal y. So by the Transitive Property they are equal to each other. 4x – 6 = x + 3

5. Systems of Equations 7-4 Additional Example 1A Continued To find y, substitute 3 for x in one of the original equations. y = x + 3 = 3 + 3 = 6 The solution is (3, 6). Solve the equation to find x. 4x – 6 = x + 3 – x – xSubtract x from both sides. 3x – 6 = 3 3x 9  6 Add 6 to both sides. 3  = 3 x = 3 Divide both sides by 3.

6. Systems of Equations 7-4 The system of equations has no solution. 2x + 9 = –8 + 2x – 2x – 2x Transitive Property Subtract 2x from both sides. 9 ≠ –8 Additional Example 1B: Solving Systems of Equations y = 2x + 9 y = –8 + 2x

7. Systems of Equations 7-4 Check It Out: Example 1A Solve the system of equations. y = x – 5 y = 2x – 8 y = x – 5y = 2x – 8 x – 5 = 2x – 8 The expressions x – 5 and 2x – 8 both equal y. So by the Transitive Property they equal each other.

8. Systems of Equations 7-4 Check It Out: Example 1A Continued To find y, substitute 3 for x in one of the original equations. y = x – 5 = 3 – 5 = –2 The solution is (3, –2). Solve the equation to find x. x – 5 = 2x – 8 – x Subtract x from both sides. –5 = x – 8 3 = x + 8 Add 8 to both sides.

9. Systems of Equations 7-4 To solve a general system of two equations with two variables, you can solve both equations for x or both for y.

10. Systems of Equations 7-4 Additional Example 2A: Solving Systems of Equations by Solving for a Variable Solve the system of equations. 5x + y = 7 x – 3y = 11 – y – y  3y  3y Solve both equations for x. 5x = 7 – y x = 11 + 3y – 15y – 15y 55 = 7 – 16y Subtract 15y from both sides. 5(11 + 3y)= 7 – y 55 + 15y = 7 – y

11. Systems of Equations 7-4 Additional Example 2A Continued –7 48 – 16y Subtract 7 from both sides. Divide both sides by –16. –16 = – 16 x = 11 + 3y = 11 + 3(–3)Substitute –3 for y. = 11 + –9 = 2 The solution is (2, –3). 55 = 7 – 16y –3 = y

12. Systems of Equations 7-4 You can solve for either variable. It is usually easiest to solve for a variable that has a coefficient of 1. Helpful Hint

13. Systems of Equations 7-4 Additional Example 2B: Solving Systems of Equations by Solving for a Variable Solve the system of equations. –2x + 10y = –8 x – 5y = 4 –10y –10y +5y +5y Solve both equations for x. –2x = –8 – 10y x = 4 + 5y = – –8 –2 10y –2 –2x –2 x = 4 + 5y 4 + 5y = 4 + 5y – 5y Subtract 5y from both sides. 4 = 4 Since 4 = 4 is always true, the system of equations has an infinite number of solutions.

14. Systems of Equations 7-4 Check It Out: Example 2A Solve the system of equations. x + y = 5 3x + y = –1 –x –x – 3x – 3x Solve both equations for y. y = 5 – x y = –1 – 3x 5 – x = –1 – 3x + x 5 = –1 – 2x Add x to both sides.

15. Systems of Equations 7-4 Check It Out: Example 2A Continued 5 = –1 – 2x + 1 6 = –2x Add 1 to both sides. Divide both sides by –2. –3 = x y = 5 – x = 5 – (–3)Substitute –3 for x. = 5 + 3 = 8 The solution is (–3, 8).

16. Systems of Equations 7-4 Check It Out: Example 2B Solve the system of equations. x + y = –2 –3x + y = 2 – x – x + 3x + 3x Solve both equations for y. y = –2 – x y = 2 + 3x –2 – x = 2 + 3x

17. Systems of Equations 7-4 + x Add x to both sides. –2 = 2 + 4x –2 –4 = 4x –2 – x = 2 + 3x Subtract 2 from both sides. Divide both sides by 4. –1 = x y = 2 + 3x = 2 + 3(–1) = –1 Substitute –1 for x. The solution is (–1, –1). Check It Out: Example 2B Continued

18. Systems of Equations 7-4 Lesson Quiz Solve each system of equations. 1. y = 5x + 10 y = –7 + 5x 2. y = 2x + 1 y = 4x 3. 6x – y = –15 2x + 3y = 5 4. Two numbers have a sum of 23 and a difference of 7. Find the two numbers. (–2, 3) 15 and 8 (, 2 ) 1 2 no solution