Systems of Equations & Inequalities

1. Systems of Equations & Inequalities Algebra I

2. Systems of Equations Definition Ways to solve • means two or more linear equations • If these two linear equations intersect, that point of intersection is called the solution • Graphing • By hand • On calculator • Substitution • Elimination

3. Graphing Method This method of solving equations is by graphing each equation on a coordinate graph. The coordinates of the intersection will be the solution to the system.

4. Graphing System of linear equations with calculator:the following two lines is a system: y=x+1 y=2x What is the solution? The point (1,2) is where the two lines intersect.

5. Practice Problem:Use Calculator:The following two lines is a system: y = 2x+1 y = 4x - 1What is the solution? The solution of this system is the point of intersection : (1,3)

6. The point where the two lines cross (4, 0) is the solution of the system. Graphing System of linear equations by hand ~ Using Table: X = 4 + y x – 3y = 4 ~ Find three values for x and y that satisfy each equation. x = 4 + y x – 3y = 4 ~ Graph these points and draw straight lines

7. Solve the following system of linear equations graphing the system by hand using slope intercept form: • The point where the two lines cross (1, 3) is the solution of the system. 2y = 4x + 2 2y = -x + 7 ~ Get the equations in slope intercept form , so you have y-intercept and slope to graph line: 2y = 4x + 2 y = (4x + 2) / 2 y = 2x +1 2y = 8x - 2 y = (8x - 2) / 2 y = 4x – 1

8. Graphing and getting Parallel Lines • If the lines are parallel • they do not intersect • there is no solution to that system.

9. Substitution Method Sometimes a system is more easily solved by the substitution method. This method involves substituting one equation into another.

10. Now insert y = 10 in one of the original equations. • x = y + 8 • x = 10 + 8 • x = 18 • Solution: • x = 18 , y = 10 • (18, 10) • From the first equation, substitute ( y + 8 ) for x in the second equation. • (y + 8) + 3y = 48 • Now solve for y. Simplify by combining y's • y + 8 + 3y = 48 • 4y + 8 = 48 • 4y = 48 – 8 • 4y = 40 • y = 40/4 • y = 10 x = y + 8 x + 3y = 48 If an equation is not already solved for one variable then you need to solve for either x or y in order to substitute.

11. Now insert x = - 4 in one of the original equations. • y = 2x + 1 • y = 2(-4) + 1 • y = -8 + 1 • y = -7 • Solution: • x = -4 , y = -7 • (-4, -7) • From the first equation, substitute ( 2x + 1 ) for y in the second equation. • 2(2x + 1) = 3x - 2 • Now solve for x: • 4x + 2 = 3x - 2 • 4x – 3x + 2 = - 2 • x + 2 = - 2 • x = - 2 – 2 • x = - 4 y = 2x + 12y = 3x - 2 If an equation is not already solved for one variable then you need to solve for either x or y in order to substitute.

12. Substitute 4 for x in either equation and solve for y: • 4 + y = 11 • y = 11 - 4 • y = 7 • Solution: • x = 4 , y = 7 • (4, 7) • From the first equation, for y: • x + y = 11 • y = 11 – x • Substitute 11 - x for y in the second equation: • 3x – (11 – x) = 5 • Now solve for x. Simplify by combining x‘s: • 3x - 11 + x = 5 • 4x = 5 + 11 • 4x = 16 • x = 16 / 4 • x = 4 x + y = 113x - y = 5 Solve the first equation for either x or y Need one variable equaling an expression so that you can substitute

13. Substitute – 6 for x in either equation and solve for y: • - 6 + y = -12 • y = -12 + 6 • y = - 6 • Solution: • x = - 6 , y = - 6 • (-6, -6) • From the second equation, for y: • x + y = -12 • y = -12 – x • Substitute -12 - x for y in the second equation: • 2x – 3(-12 – x) = 6 • Now solve for x: • 2x + 36 + 3x = 6 • 5x + 36 = 6 • 5x = 6 - 36 • 5x = - 30 • x = - 30 / 5 • x = - 6 2x – 3y = 6x + y = -12 Solve the first equation for either x or y Need one variable equaling an expression so that you can substitute

14. Individual Practice Problems: Substitution Method • Problem 1: • y= x + 1 • 2y= 3x • Problem 2: • y – 5x = - 1 • 2y= 3x + 12 • Problem 3: • y = 3x + 1 • 4y = 12x + 4 • Problem 4: • y – 3x = 1 • 4y = 12x + 3

15. Answers to Practice Problems • Problem 1: • y= x + 1 • 2y= 3x

16. Answers to Practice Problems • Problem 2: y – 5x = - 1 2y= 3x + 12 • Solve 1st equation for y: • y = 5x - 1

17. Answers to Practice Problems • Problem 3: • y = 3x + 1 • 4y = 12x + 4

18. Answers to Practice Problems • Problem 4: y – 3x = 1 4y = 12x + 3 • Solve 1st equation for y: • y = 3x + 1

19. Elimination Method Helps eliminate one variable so that you can solve for the remaining variable.

20. Steps for Elimination Method • Multiply one or both equations by some number to make the number in front of one of the letters (unknowns) the same in each equation. • Add or subtract the two equations to eliminate one letter. • Solve for the other unknown. • Insert the value of the first unknown in one of the original equations to solve for the second unknown.

21. x + y = 7 • + x – y = 3 2x = 10 • solve for x: • 2x = 10 • x = 10 / 2 • x = 5 • replace x value and find y: • 5 + y = 7 • y = 7 – 5 • y = 2 x + y = 7x – y = 3 ~ You have opposites between the two equations a positive and negative y ~ With opposites you can skip step one and add equations together • Solution: • x = 5 and y = 2 • (5, 2)

22. Need to get the same y’s in both: • change 2nd equation to have 3y • 3(2x + y) = 3(13) • 6x + 3y = 39 • 3x + 3y = 24 • - 6x + 2y = 39 -3x = -15 • solve for x: • -3x = -15 • x = -15 / -3 • x = 5 • replace x value and find y: • 2x + y = 13 • 2(5) + y = 13 • 10 + y = 13 • y = 13 - 10 • y = 3 3x + 3y = 242x + y = 13 ~ You need to have the same variable in both equations… ~ With the same variable you can subtract equations together • Solution: • x = 5 and y = 2 • (5, 2)

23. Multiply the second equation by 5 to make the x-coefficient a multiple of 5: • 5(3x - 5y) = 5(-23) • 15x - 25y = -115 • multiply the first equation by 3, to get the same x-coefficient: • 3(5x + 3y) = 3(7) • 15x + 9y = 21 • 15x - 25y = -115 • -15x + 9y = 21 -34y = -136 • solve for y: • -34y = -136 • y = -136 / -34 • y = 4 • replace 4 for y in one of the original equations and find x: • 5x + 3y = 7 • 5x + 3(4)= 7 • 5x + 12 = 7 • 5x = 7 - 12 • 5x = -5 • x = -5 / 5 • x = -1 5x + 3y = 73x - 5y = -23 ~ You need to have the same variable in both equations… ~ With the same variable you can subtract equations together • Solution: • x = -1 & y = 4 • (-1, 4)

24. Individual Practice Problems: Elimination Method • Problem 1: • y = x + 1  • y = –x • Problem 2: • 4x – 2y = 14 • x + 2y = 6 • Problem 3: • 2x + 3y = 17 • 4x – 3y = 1 • Problem 4: • y = 2x + 1 • y = -4x + 1

25. Answers to Practice Problems • Problem 1: • y = x + 1  • y = –x

26. Answers to Practice Problems • Problem 2: • 4x – 2y = 14 • x + 2y = 6

27. Answers to Practice Problems • Problem 3: • 2x + 3y = 17 • 4x – 3y = 1

28. Answers to Practice Problems • Problem 4: • y = 2x + 1 • y = -4x + 1 • Need to get same coefficient fro the x variables

29. Story Problemswith Systems of Equations

30. Steps to Solve • Define the variables • Set up the equations • Solve the system • Graphing • Substitution • Elimination • Put answer back into form of the problem

31. The admission fee at a small fair is $1.50 for children and $4.00 for adults. On a certain day, 2200 people enter the fair and $5050 is collected. How many children and how many adults attended? • Define Variables: • number of adults: anumber of children: c • Write Equations: • total number:  a + c = 2200 total income:  4a + 1.5c = 5050 • Solve for variables: • Solve 1st equation for a: • a = 2200 – c • Use (2200 – c) for a in 2nd to find c: • 4(2200 – c) + 1.5c = 5050 • 8800 – 4c + 1.5c = 5050  • 8800 – 2.5c = 5050  • -2.5c = 5050 - 8800 • –2.5c = –3750  • c = -3750 / -2.5 • c = 1500 • Use 1500 for c to find a: • a = 2200 – (1500) = 700 • Write out answer • There were 1500 children and 700 adults.

32. A landscaping company placed two orders with a nursery. The first order was for 13bushes and 4 trees, and totaled $487. The second order was for 6 bushes and 2 trees, and totaled $232. The bills do not list the per-item price. What were the costs of one bush and of one tree? • Define Variables: • b = the number of bushes • t = the number of trees • Write Equations: • first order: 13b + 4t = 487 second order: 6b + 2t = 232 • Solve for variables: • use elimination • Multiplying the second by –2 • -2 (6b + 2t) = -2(232) • -12b – 4t = -464 • 13b + 4t = 487 + –12b – 4t = –464 • b = 23 • Use 23 for b to find t: • 13b + 4t = 487 • 13(23) + 4t = 487 • 299 + 4t = 487 • 4t = 487 – 299 • 4t = 188 • t = 188 / 4 • t = 47 • Write out answer • Bushes cost $23 each; trees cost $47 each.

33. The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number. • Define Variables: • t = tens digit of the original number • u = units (or "ones") digit • Write Equations: • 1st Sentence gives you: • t + u = 7 • Think about numbers to get original numbers: • 26 is 10 times 2, plus 6 times 1 • two-digit number will be ten times the (tens digit), plus one times the (units digit) • original number: 10t + 1u • New number has the digits reversed:(switch the place of t & u)new number: 10u + 1t • (new number) is (old number) increased by (twenty-seven) • 10u + 1t = 10t + 1u + 27

34. The sum of the digits of a two-digit number is 7. When the digits are reversed, the number is increased by 27. Find the number. ~ Continued • System to solve: • t + u = 7  • 10u + t = 10t + u + 27 • Simplify the second equation: • 10u + t = 10t + u + 27 • 10u – u + t – 10t = 27 • 9u – 9t = 27  • 9(u – t) = 27 • u – t = 27 / 9 • u – t = 3 • Write to look likeother: • -t + u = 3 • Solve for variables: • Use elimination! • t + u = 7 • + -t + u = 3 • 2u = 10 • u = 10 / 2 • u = 5 • Use 5 for u to find t: • t + u = 7 • t + 5 = 7 • t = 7 – 5 • t = 2 • Write out answer • The number is 25.

35. A passenger jet took three hours to fly 1800 miles in the direction of the jetstream. The return trip against the jetstream took four hours. What was the jet's speed in still air and the jetstream's speed? • Define Variables: • p = the plane's speedometer reading • w = the windspeed • Write Equations: • the plane is going "with" the wind the two speeds will be added together • when the plane is going "against" the wind the windspeed will be subtracted from the plane's speedometer reading • the distance equation will be: • (the combined speed) times (the time at that speed) equals (the total distance travelled) • with the jetstream: against the jetstream: •  (p + w)(3) = 1800 (p – w)(4) = 1800 • p + w = 1800 / 3 p – w = 1800 / 4 • p + w = 600 p – w = 450 

36. A passenger jet took three hours to fly 1800 miles in the direction of the jetstream. The return trip against the jetstream took four hours. What was the jet's speed in still air and the jetstream's speed? • Solve for variables: • use elimination Method • p + w = 600 + p – w = 450 • 2p = 1050 • p = 1050 / 2 • p = 525 • Use 525 for p to find w: • p + w = 600 • 525 + w = 600 • w = 600 – 525 • w = 75 • Write out answer • The jet's speed was 525 mph and the jetstreamwindspeed was 75 mph.

37. Systems of Inequalities

38. Solving Systems of Linear Inequalities • The solution to a system of linear inequalities is shown by graphing them. • Need to put the inequalities into Slope-Intercept Form, y = mx + b.

39. Solving Systems of Linear Inequalities • Lines on the graph • If the inequality is < or >, make the lines dotted. • If the inequality is < or >, make the lines solid.

40. Solving Systems of Linear Inequalities • The solution also includes points not on the line, so you need to shade the region of the graph: • above the line for ‘y >’ or ‘y ’. • below the line for ‘y <’ or ‘y ≤’.

41. Solving Systems of Linear Inequalities Example: a: 3x + 4y > - 4 b: x + 2y < 2 Put in Slope-Intercept Form:

42. a: dotted shade above b: dotted shade below Solving Systems of Linear Inequalities Example, continued: Graph each line, make dotted or solid and shade the correct area.

43. Solving Systems of Linear Inequalities a: 3x + 4y > - 4

44. Solving Systems of Linear Inequalities a: 3x + 4y > - 4 b: x + 2y < 2

45. Solving Systems of Linear Inequalities a: 3x + 4y > - 4 b: x + 2y < 2 The area between the green arrows is the region of overlap and thus the solution.