Learn to solve systems of equations .

1. Learn to solve systems of equations.

2. 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. Caution! When solving systems of equations, remember to find values for all of the variables.

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

5. Additional Example 1A Continued Solve the equation to find x. 4x – 6 = x + 3 – x– x Subtract x from both sides. 3x – 6 = 3 + 6+ 6 Add 6 to both sides. 3x9 Divide both sides by 3. 3 = 3 x = 3 To find y, substitute 3 for x in one of the original equations. y = x + 3 = 3 + 3 = 6 The solution is (3, 6).

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

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

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

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

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

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

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

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

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

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

16. Check It Out: Example 2B Solve the system of equations. x + y = –2 –3x + y = 2 Solve both equations for y. x + y = –2 –3x + y = 2 – x– x+ 3x+ 3x y = –2 – xy = 2 + 3x –2 – x = 2 + 3x

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

18. 1 2 ( , 2) 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. no solution (–2, 3) 15 and 8