Linear Systems and Problem Solving

Ways to Solve a System of Linear Equations Graphing – can provide a useful method for estimating a solution and to provide a visual model of the problem. Substitution – requires that one of the variables be isolated on one side of the equation. It is especially convenient when one of the variables has a coefficient of 1 or –1. Elimination Using Addition –convenient when a variable appears in different equations with coefficients that are opposites. Elimination Using Subtraction –convenient if one of the variables has the same coefficient in the two equations. Elimination Using Multiplication –can be applied to create opposites in any system.

Solving Word Problems Using A Linear System 1) Write two sets of labels, if necessary (one set for number, one set for value, weight etc.) 2) Write two verbal models. (Translate from sentences) 3) Write two algebraic models (equations). 4) Solve the linear system. 5) Write a sentence and check your solution in the word problem.

Meg’s age is 5 times Jose’s age. The sum of their ages is 18 Meg’s age is 5 times Jose’s age. The sum of their ages is 18. How old is each person? Assign Labels. Choose a different variable for each person. Let m = Meg’s age Let j = Jose’s age Write an equation for each of the first two sentences. m = 5j m + j = 18 Solve the system of equations. How old is Meg? Sentence. Jose is 3 and Meg is 15.

The length of a rectangle is 1 m more than twice its width The length of a rectangle is 1 m more than twice its width. If the perimeter is 110 m, find the dimensions. = length let l = length let w = width width width length Formula The width is 18 m and the length is 37 m.

Example 1 A class has a total of 25 students Example 1 A class has a total of 25 students. Twice the number of girls is equal to 3 times the number of boys. How many boys and girls are there in the class? Assign Labels. Choose a different variable for each type of person. Let g = # of girls Let b = # of boys Write an equation for each of the first two sentences. g + b = 25 2g = 3b There are 15 girls and 10 boys in the class.

Example 2 The length of a rectangle is 4 m more than twice its width Example 2 The length of a rectangle is 4 m more than twice its width. If the perimeter is 38 m, find the dimensions. = length 1. Labels. let w = width let l = length width width length 2. Translate first sentence. 3. Use perimeter formula. 4. Solve the system. 5. Sentence. The width is 5 m and the length is 14 m.

Example 3 Admission to the play was $2 for an adult and $1 Example 3 Admission to the play was $2 for an adult and $1.50 for a student. Total income from the sale of tickets was $550. The number of adult tickets sold was 100 less than 3 times the number of student tickets. How many tickets of each type were sold? Number Labels. let a = # of adult tickets let s = # of student tickets value of adult tickets value of student tickets Value Labels. let 2a = let 1.50s =

Example 3 Admission to the play was $2 for an adult and $1 Example 3 Admission to the play was $2 for an adult and $1.50 for a student. Total income from the sale of tickets was $550. The number of adult tickets sold was 100 less than 3 times the number of student tickets. How many tickets of each type were sold? = Number Labels. let a = # of adult tickets let s = # of student tickets value of adult tickets value of student tickets Value Labels. let 2a = let 1.50s = a = 3s – 100 2a + 1.50s Clear the decimals. Multiply both sides by 100. = 550 The school sold 200 adult tickets and 100 student tickets.

Example 4 The number of quarters that Tom has is 3 times the number of nickels. He has $1.60 in all. How many coins of each type does he have? Number Labels. let q = # of quarters let n = # of nickels Value Labels. let .25q = value of quarters let .05n = value of nickels

Tom has 6 quarters and 2 nickels. Example 4 The number of quarters that Tom has is 3 times the number of nickels. He has $1.60 in all. How many coins of each type does he have? = Number Labels. let q = # of quarters let n = # of nickels Value Labels. let .25q = value of quarters let .05n = value of nickels q = 3n .25q + .05n = 1.60 Clear the decimals. Multiply both sides by 100. Tom has 6 quarters and 2 nickels.

Example 5 The sum of two numbers is 100 Example 5 The sum of two numbers is 100. Five times the smaller number is 8 more than the larger number. What are the two numbers? Assign Labels. Let s = smaller # Let l = larger # Equations. s + l = 100 5s = l + 8 The larger number is 82 and the smaller number is 18.

Example 6 One number is 12 more than half another number Example 6 One number is 12 more than half another number. The two numbers have a sum of 60. Find the numbers. Assign Labels. Let x = first # Let y = second # Equations. One number is 28 and the other number is 32.

Example 7 If you buy six pens and one mechanical pencil, you’ll get $1 change from your $10 bill. But if you buy four pens and two mechanical pencils, you’ll get $2 change. How much does each pen and pencil cost? Assign Labels. Let p = pens Let m = mechanical pencils 6p + m = 10 - 1 Equations. 6p + m = 10 - 1 4p + 2m = 10 - 2 Pens cost $1.25 each and mechanical pencils cost $1.50 each.

SOLVE THE WORD PROBLEM: An exam worth 145 points contains 50 questions. Some of the questions are worth two points and some are worth five points. How many two point questions are on the test? How many five point questions are on the test?

Let x = the number of 2 point questions and An exam worth 145 points contains 50 questions. Some of the questions are worth two points and some are worth five points. How many two point questions are on the test? How many five point questions are on the test? DEFINE THE VARIABLES: Let x = the number of 2 point questions and y = the number of 5 point questions. WRITE A SYSTEM OF EQUATIONS:

SOLVE FOR ONE VARIABLE: x = 35 two-point questions An exam worth 145 points contains 50 questions. Some of the questions are worth two points and some are worth five points. How many two point questions are on the test? How many five point questions are on the test? SOLVE FOR ONE VARIABLE: x = 35 two-point questions SOLVE FOR THE OTHER VARIABLE: x + y = 50 35 + y = 50 y = 15 five-point questions

CHECK THE SOLUTION: (35, 15) 2(35) + 5(15) = 145 70 + 75 = 145 An exam worth 145 points contains 50 questions. Some of the questions are worth two points and some are worth five points. How many two point questions are on the test? How many five point questions are on the test? CHECK THE SOLUTION: (35, 15) 2(35) + 5(15) = 145 70 + 75 = 145 145 = 145  35 + 15 = 50 50 = 50 

SOLVE THE WORD PROBLEM: The Lakers scored a total of 80 points in a basketball game against the Bulls. The Lakers made a total of 37 two-point and three-point baskets. How many two-point shots did the Lakers make? How many three-point shots did the Lakers make?

Let x = the number of 2 point baskets and The Lakers scored a total of 80 points in a basketball game against the Bulls. The Lakers made a total of 37 two-point and three-point baskets. How many two-point shots did the Lakers make? How many three-point shots did the Lakers make? DEFINE THE VARIABLES: Let x = the number of 2 point baskets and y = the number of 3 point baskets. WRITE A SYSTEM OF EQUATIONS:

SOLVE FOR ONE VARIABLE: x = 31 two-point shots The Lakers scored a total of 80 points in a basketball game against the Bulls. The Lakers made a total of 37 two-point and three-point baskets. How many two-point shots did the Lakers make? How many three-point shots did the Lakers make? SOLVE FOR ONE VARIABLE: x = 31 two-point shots SOLVE FOR THE OTHER VARIABLE: x + y = 37 31 + y = 37 y = 6 three-point shots

CHECK THE SOLUTION: (31, 6) 2(31) + 3(6) = 80 62 + 18 = 80 80 = 80  The Lakers scored a total of 80 points in a basketball game against the Bulls. The Lakers made a total of 37 two-point and three-point baskets. How many two-point shots did the Lakers make? How many three-point shots did the Lakers make? CHECK THE SOLUTION: (31, 6) 2(31) + 3(6) = 80 62 + 18 = 80 80 = 80  x + y = 37 31 + 6 = 37 37 = 37 

SOLVE THE WORD PROBLEM: Next week your math teacher is giving a chapter test worth 100 points. The test will consist of 35 problems. Some problems are worth 2 points and some problems are worth 4 points. How many problems of each value are on the test?

Let x = the number of 2 point problems and Next week your math teacher is giving a chapter test worth 100 points. The test will consist of 35 problems. Some problems are worth 2 points and some problems are worth 4 points. How many problems of each value are on the test? DEFINE THE VARIABLES: Let x = the number of 2 point problems and y = the number of 4 point problem. WRITE A SYSTEM OF EQUATIONS:

SOLVE FOR ONE VARIABLE: x = 20 two-point problems Next week your math teacher is giving a chapter test worth 100 points. The test will consist of 35 problems. Some problems are worth 2 points and some problems are worth 4 points. How many problems of each value are on the test? SOLVE FOR ONE VARIABLE: x = 20 two-point problems SOLVE FOR THE OTHER VARIABLE: x + y = 35 20 + y = 35 y = 15 three-point problems

CHECK THE SOLUTION: (20, 15) 2(20) + 4(15) = 100 40 + 60 = 100 Next week your math teacher is giving a chapter test worth 100 points. The test will consist of 35 problems. Some problems are worth 2 points and some problems are worth 4 points. How many problems of each value are on the test? CHECK THE SOLUTION: (20, 15) 2(20) + 4(15) = 100 40 + 60 = 100 100 = 100  x + y = 35 20 + 15 = 35 35 = 35 

SOLVE THE WORD PROBLEM: You are selling tickets for a musical at your local community college.  Student tickets cost $5 and general admission tickets cost $8. If you sell 500 tickets and collect $3475, how many student tickets and how many general admission?

Let x = the number of student tickets and You are selling tickets for a musical at your local community college.  Student tickets cost $5 and general admission tickets cost $8. If you sell 500 tickets and collect $3475, how many student tickets and how many general admission? DEFINE THE VARIABLES: Let x = the number of student tickets and y = the number of general tickets. WRITE A SYSTEM OF EQUATIONS:

SOLVE FOR ONE VARIABLE: x = 175 student tickets You are selling tickets for a musical at your local community college.  Student tickets cost $5 and general admission tickets cost $8. If you sell 500 tickets and collect $3475, how many student tickets and how many general admission? SOLVE FOR ONE VARIABLE: x = 175 student tickets SOLVE FOR THE OTHER VARIABLE: x + y = 500 175 + y = 500 y = 325 general tickets

CHECK THE SOLUTION: (175, 325) 5(175) + 8(325) = 3475 You are selling tickets for a musical at your local community college.  Student tickets cost $5 and general admission tickets cost $8. If you sell 500 tickets and collect $3475, how many student tickets and how many general admission? CHECK THE SOLUTION: (175, 325) 5(175) + 8(325) = 3475 875 + 2600 = 3475 3475 = 3475  x + y = 500 175 + 325 = 500 500 = 500 

SOLVE THE WORD PROBLEM: The Madison Local High School marching band sold gift wrap to earn money for a band trip to Orlando, Florida. The gift wrap in solid colors sold for $4.00 per roll and the print gift wrap sold for $6.00 per roll. The total number of rolls sold was 480 and the total amount of money collected was $2340. How many rolls of each kind of gift wrap were sold?

Let x = the amount of solid rolls and y = the amount of printed rolls. The Madison Local High School marching band sold gift wrap to earn money for a band trip to Orlando, Florida. The gift wrap in solid colors sold for $4.00 per roll and the print gift wrap sold for $6.00 per roll. The total number of rolls sold was 480 and the total amount of money collected was $2340. How many rolls of each kind of gift wrap were sold? DEFINE THE VARIABLES: Let x = the amount of solid rolls and y = the amount of printed rolls. WRITE A SYSTEM OF EQUATIONS:

SOLVE FOR ONE VARIABLE: x = 270 solid rolls The Madison Local High School marching band sold gift wrap to earn money for a band trip to Orlando, Florida. The gift wrap in solid colors sold for $4.00 per roll and the print gift wrap sold for $6.00 per roll. The total number of rolls sold was 480 and the total amount of money collected was $2340. How many rolls of each kind of gift wrap were sold? SOLVE FOR ONE VARIABLE: x = 270 solid rolls SOLVE FOR THE OTHER VARIABLE: x + y = 480 270 + y = 480 y = 210 printed rolls

CHECK THE SOLUTION: (270, 210) 4(270) + 6(210) = 2340 The Madison Local High School marching band sold gift wrap to earn money for a band trip to Orlando, Florida. The gift wrap in solid colors sold for $4.00 per roll and the print gift wrap sold for $6.00 per roll. The total number of rolls sold was 480 and the total amount of money collected was $2340. How many rolls of each kind of gift wrap were sold? CHECK THE SOLUTION: (270, 210) 4(270) + 6(210) = 2340 1080 + 1260 = 2340 2340 = 2340  x + y = 480 270 + 210 = 480 480 = 480 