1. Section 7.2 Systems of Linear Equations Math in Our World

2. Learning Objectives  Solve systems of linear equations graphically.  Identify inconsistent and dependent systems.  Solve systems of linear equations by substitution.  Solve systems of linear equations by the addition/subtraction method.  Solve real-world problems using systems of linear equations.

3. System of Equations A system of two linear equations in two variables x and y is a pair of equations that can be written in the form Solving a system of two linear equations means finding all pairs of numbers that satisfy both equations in the system.

4. Solving a System Graphically Solving a System of Equations Graphically Step 1 Draw the graphs of both equations on the same Cartesian plane. Step 2 Find the point or points of intersection, if there are any. We know that the graph of any linear equation with two variables is a straight line. We also know that the coordinates of every point on the line satisfy the equation. So a pair of numbers that satisfies both equations in a system must correspond to a point that is on both lines.

5. EXAMPLE 1 Solving a System Graphically Solve the system graphically:

6. EXAMPLE 1 Solving a System Graphically SOLUTION Step 1 Graph both equations. Step 2 Find the point of intersection. In this case, it is (5, 1), so the solution set is {(5, 1)}.

7. Three Possibilities with Systems 1. The lines intersect at a single point. In this case, there is only one solution, and that is the point of intersection of the lines. This system is said to be consistent and independent.

8. Three Possibilities with Systems 2. The lines are parallel. In this case, there would be no solution, or the solution would be the empty set, , since parallel lines never intersect. This system is said to be inconsistent and independent.

9. Three Possibilities with Systems 3. The lines coincide. In this case, the graph of both equations is the same line; so any point on the line will satisfy both equations. There are infinitely many solutions. This system is said to be consistent and dependent. The solution set is written as {(x, y) | ax + by = c}, where ax + by = c is either one of the two equations.

10. EXAMPLE 2 Identifying an Inconsistent System Graphically Solve the system graphically:

11. EXAMPLE 2 Identifying an Inconsistent System Graphically SOLUTION Step 1 Graph both equations. Step 2 Find the point or points of intersection. In this case, the lines are parallel, and never intersect. The solution set is , and the system is inconsistent.

12. EXAMPLE 3 Identifying a Dependent System Graphically Solve the system graphically:

13. EXAMPLE 3 Identifying a Dependent System Graphically SOLUTION Step 1 Graph both equations. Step 2 Point(s) of intersection. In this case, the lines coincide; so the solution set is any point on either line. We can write this as {(x, y) | x – 5y = 15}. The system is dependent.

14. EXAMPLE 4 An Application of the Graphical Method Mary leaves home at 9 A.M. and starts hiking around a lake at 3 miles per hour. At 10 A.M., her boyfriend Dave follows her, jogging at 5 miles per hour. When and how far from home do they meet? SOLUTION There are two variables for each person: the amount of time passed, and how far they’ve gone. We will choose 10 A.M. as our base time, since they’re both moving after that point. Let x = the time in hours after 10 A.M., and let y = the distance traveled.

15. EXAMPLE 4 An Application of the Graphical Method SOLUTION Mary: At 10 A.M., she’s been hiking for an hour, and has gone 3 miles (her speed is 3 miles/hour). Her rate of change (slope) is 3, so Mary’s distance is y = 3x + 3. Dave: At 10 A.M., his distance is zero, and his rate of change is 5, so Dave’s distance is y = 5x. Now we have the system

16. EXAMPLE 4 An Application of the Graphical Method SOLUTION The graphs of the two equations shown to the right intersect at (1.5, 7.5), so Dave and Mary are at the same place at the same time 1.5 hours after 10 A.M., which is 11:30. And at that time, they are 7.5 miles from home.

17. Solving a System by Substitution Solving a System of Equations by Substitution Step 1 Pick one equation and solve it for one variable (either x or y) in terms of the other variable. Step 2 Substitute the expression that you found in step 1 into the other equation for the variable you solved for. Step 3 Solve the resulting equation for the unknown (it now has only one variable). Step 4 Pick one of the original equations, substitute the value found in step 3 for the variable, and solve to find the value of the other variable.

18. EXAMPLE 5 Solving a System by Substitution Solve the system: SOLUTION Step 1 It will be easy to solve the first equation for x, so we start there. Step 2 Substitute the expression (8 – 3y) for x into the second equation.

19. EXAMPLE 5 Solving a System by Substitution SOLUTION Step 3 Solve the equation for y.

20. EXAMPLE 5 Solving a System by Substitution SOLUTION Step 4 Substitute y = 1 into either equation and solve for x. The solution set is {(5, 1)}. Note that this is the same solution obtained by graphing, as shown in Example 1.

21. EXAMPLE 6 Solving a System by Substitution Solve the system: SOLUTION Step 1 Neither of the variables in either equation is easier to solve for than the other, so we randomly choose to solve the first equation for x.

22. EXAMPLE 6 Solving a System by Substitution SOLUTION Step 2 Substitute the expression for x into the second equation.

23. EXAMPLE 6 Solving a System by Substitution SOLUTION Step 3 Solve the equation for y.

24. EXAMPLE 6 Solving a System by Substitution SOLUTION Step 4 Substitute y = – 1 into one equation and solve for x. The solution set is {(2, –1)}.

25. Solving a System by Addition Solving a System of Equations by Addition Method Step 1 If necessary, rewrite the equations so they are in the form ax + by = c. Step 2 Multiply one or both of the equations by a number so that the two coefficients of either x or y have the same absolute value. Step 3 Either add or subtract the two equations so that one of the variables is eliminated. Step 4 Solve the resulting single-variable equation for the variable. Step 5 Substitute the value of the variable from Step 4 into either of the original equations, and solve for the other variable.

26. EXAMPLE 7 Solving a System by Addition/Subtraction Solve the system: SOLUTION Step 1 Write both equations in the form ax + by = c. (Add y to both sides of the second equation.)

27. EXAMPLE 7 Solving a System by Addition/Subtraction SOLUTION Step 2 Multiply the second equation by 2 in order to make the coefficients of the x terms equal. which gives us

28. EXAMPLE 7 Solving a System by Addition/Subtraction SOLUTION Step 3 Subtract the second equation from the first equation to eliminate the x variable. Step 4 Solve the equation for y.

29. EXAMPLE 7 Solving a System by Addition/Subtraction SOLUTION Step 5 Pick one equation and substitute 4 for y, then solve for x. The solution is (3, 4).

30. EXAMPLE 8 Solving a System by Addition/Subtraction Solve the system: SOLUTION Step 1 Write both equations in the form ax + by = c. (Subtract 56 from both sides of the second equation.)

31. EXAMPLE 8 Solving a System by Addition/Subtraction SOLUTION Step 2 Multiply the first equation by 5 and the second equation by 2 to make the coefficients of the y terms equal in absolute value. which gives us

32. EXAMPLE 8 Solving a System by Addition/Subtraction SOLUTION Step 3 Add the equations. Step 4 Solve the equation for x.

33. EXAMPLE 8 Solving a System by Addition/Subtraction SOLUTION Step 5 Pick one equation and substitute – 2 for x, then solve for x. The solution is (– 2, – 10).

34. EXAMPLE 9 Identifying an Inconsistent System Solve the system:

35. EXAMPLE 9 Identifying an Inconsistent System SOLUTION Solving by substitution, we get Since the resulting equation, 24 = 15, is false, the system is inconsistent. The lines are parallel, and the solution set is .

36. EXAMPLE 10 Identifying a Dependent System Solve the system: SOLUTION This time, for variety, we’ll use the addition/subtraction method. First, we multiply the first equation by 4. which gives us

37. EXAMPLE 10 Identifying a Dependent System SOLUTION Now we add the equations Since the variables are all eliminated and the resulting equation is true, the system is dependent. The solution set is {(x, y) | 5x + y = 9}.

38. EXAMPLE 11 Applying Systems of Equations to Finance A small business owner plans to set up an investment plan that will generate enough income to pay the property taxes on her shop, a total of $2,200 per year. She has $24,000 to invest, and divides that amount between two accounts. One pays 8% annually. The other is a bit riskier, but it pays 10% annually unless it collapses. How much should be invested in each account to raise the required $2,200 in annual income?

39. EXAMPLE 11 Applying Systems of Equations to Finance SOLUTION Let x = the amount of money invested at 8%. Let y = the amount of money invested at 10%. The first equation is x + y = $24,000 since this is the total amount of money invested. The second equation is 0.08x + 0.10y = $2,200, since this is the amount of interest earned from 8% of the investment amount x and 10% of the investment amount y. The system is

40. EXAMPLE 11 Applying Systems of Equations to Finance SOLUTION Solving the system by substitution:

41. EXAMPLE 11 Applying Systems of Equations to Finance SOLUTION $10,000 should be invested at 8%. This leaves $14,000 invested at 10%.

42. EXAMPLE 12 Applying Systems of Equations to Unit Cost A campus bookstore received two shipments from Apple Computer over the last month. The first contained six iPods and eight MacBooks, and the cost to the bookstore was $6,840. The second shipment was three iPods and five MacBooks, at a wholesale cost of $4,170. The bookstore manager is unable to find the itemized invoice, and accounting needs to know how much each individual item cost. What were the individual costs?

43. EXAMPLE 12 Applying Systems of Equations to Unit Cost SOLUTION Let x = the iPod cost. Let y = the MacBook cost. Then 6x + 8y is the cost of the first shipment, and 3x + 5y is the cost of the second. This gives us a system:

44. EXAMPLE 12 Applying Systems of Equations to Unit Cost SOLUTION We will solve using addition/subtraction. Multiply the second equation by 2: Now subtract the second equation from the first, then solve for y:

45. EXAMPLE 12 Applying Systems of Equations to Unit Cost SOLUTION The bookstore paid $140 for each iPod and $750 for each MacBook. Substitute y = 750 into one of the original equations and solve for x: