Please use attached file

1. ?/1 pointsBerrFinMath1 3.1.001. Solve the system by graphing. (Enter your answers as a commaseparated list. If the system is inconsistent, enter INCONSISTENT. If

the system is dependent, enter DEPENDENT.) x + y = 10

x?y=4 (x, y) =

Show My Work (Optional)

2. ?/1 pointsBerrFinMath1 3.1.004. Solve the system by the elimination method. (Enter your answers as a commaseparated list. If the system is inconsistent, enter

INCONSISTENT. If the system is dependent, enter DEPENDENT.) 3x + y = 23

x + 2y = 16 (x, y) =

Show My Work (Optional)

3. ?/1 pointsBerrFinMath1 3.1.005. Solve the system by the elimination method. (Enter your answers as a commaseparated list. If the system is inconsistent, enter

INCONSISTENT. If the system is dependent, enter DEPENDENT.) x + 2y = 16

3x + 4y = 34 (x, y) =

Show My Work (Optional)

4. ?/1 pointsBerrFinMath1 3.1.006. Solve the system by the elimination method. (Enter your answers as a commaseparated list. If the system is inconsistent, enter

INCONSISTENT. If the system is dependent, enter DEPENDENT.) 2x + 5y = 55

2x + 3y = 41 (x, y) =

Show My Work (Optional)

https://www.webassign.net/web/Student/Assignment-Responses/last?dep=13959992 1/3

7/8/2016 Unit 2 Quiz (3.1, 3.2)

5. ?/2 pointsBerrFinMath1 3.1.008. Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x and y

variables. Solve the system by the elimination method. Be sure to state your final answer in terms of the original question.

A lawyer has found 60 investors for a limited partnership to purchase an innercity apartment building, with each contributing either $3,000 or $6,000. If the partnership raised $246,000, then how many investors contributed $3,000 and how many contributed $6,000?

x = $3,000 investors y = $6,000 investors

Show My Work (Optional)

6. ?/2 pointsBerrFinMath1 3.1.009. Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x and y

variables. Solve the system by the elimination method. Be sure to state your final answer in terms of the original question.

A jar contains 80 nickels and dimes worth $7.30. How many of each kind of coin are in the jar? x = nickels y = dimes

Show My Work (Optional)

7. ?/2 pointsBerrFinMath1 3.1.010. Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x and y

variables. Solve the system by the elimination method. Be sure to state your final answer in terms of the original question.

The concession stand at an ice hockey rink had receipts of $6200 from selling a total of 2600 sodas and hot dogs. If each soda sold for $2 and each hot dog sold for $3, how many of each were sold?

x = sodas y = hot dogs

Show My Work (Optional)

8. ?/6 pointsBerrFinMath1 3.2.006. Carry out the row operation on the matrix.

R1?R2?R1 on

49 50 57 52

Show My Work (Optional)

9. ?/6 pointsBerrFinMath1 3.2.007. Carry out the row operation on the matrix.

1R2?R2 on 4 ?2 ?43 7 0770

Show My Work (Optional)

https://www.webassign.net/web/Student/Assignment-Responses/last?dep=13959992 2/3

7/8/2016 Unit 2 Quiz (3.1, 3.2)

10.?/1 pointsBerrFinMath1 3.2.008. Interpret the augmented matrix as the solution of a system of equations. (Enter your answers as a commaseparated list. If the system

is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.) 106

01 ?3 (x, y) =

Show My Work (Optional)

11.?/1 pointsBerrFinMath1 3.2.011. Solve the system by rowreducing the corresponding augmented matrix. (Enter your answers as a commaseparated list. If the system

is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.) 2x + y = 20

x + y = 14 (x, y) =

Show My Work (Optional)

12.?/2 pointsBerrFinMath1 3.2.015. Express the situation as a system of two equations in two variables. Be sure to state clearly the meaning of your x and yvariables.

Solve the system by rowreducing the corresponding augmented matrix. State your final answer in terms of the original question.

For the final days before the election, the campaign manager has a total of $47,000 to spend on TV and radio campaign advertisements. Each TV ad costs $3000 and is seen by 10,000 voters, while each radio ad costs $500 and is heard by 2000 voters. Ignoring repeated exposures to the same voter, how many TV and radio ads will contact 164,000 voters using the allocated funds?

x = TV ads y = radio ads

7/8/2016 Unit 2 Quiz (3.1, 3.2) WebAssign Unit 2 Quiz (3.1, 3.2) (Quiz) Current Score : - / 26 Ryan Manuel MAT230, section MAT230.31 MOD6 2016, Summer 2 2016 Instructor: Kathleen Ferguson Due : Sunday, July 10 2016 11:59 PM EDT 1. -/1 pointsBerrFinMath1 3.1.001. Solve the system by graphing. (Enter your answers as a commaseparated list. If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.) x + y = 10 x y = 4 (x, y) = Show My Work (Optional) 2. -/1 pointsBerrFinMath1 3.1.004. Solve the system by the elimination method. (Enter your answers as a commaseparated list. If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.) 3x + y = 23 x + 2y = 16 (x, y) = Show My Work (Optional) 3. -/1 pointsBerrFinMath1 3.1.005. Solve the system by the elimination method. (Enter your answers as a commaseparated list. If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.) x + 2y = 16 3x + 4y = 34 (x, y) = Show My Work (Optional) 4. -/1 pointsBerrFinMath1 3.1.006. Solve the system by the elimination method. (Enter your answers as a commaseparated list. If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.) 2x + 5y = 55 2x + 3y = 41 (x, y) = Show My Work (Optional) https://www.webassign.net/web/Student/Assignment-Responses/last?dep=13959992 1/3 7/8/2016 Unit 2 Quiz (3.1, 3.2) 5. -/2 pointsBerrFinMath1 3.1.008. Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x and y variables. Solve the system by the elimination method. Be sure to state your final answer in terms of the original question. A lawyer has found 60 investors for a limited partnership to purchase an innercity apartment building, with each contributing either $3,000 or $6,000. If the partnership raised $246,000, then how many investors contributed $3,000 and how many contributed $6,000? x = $3,000 investors y = $6,000 investors Show My Work (Optional) 6. -/2 pointsBerrFinMath1 3.1.009. Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x and y variables. Solve the system by the elimination method. Be sure to state your final answer in terms of the original question. A jar contains 80 nickels and dimes worth $7.30. How many of each kind of coin are in the jar? x = nickels y = dimes Show My Work (Optional) 7. -/2 pointsBerrFinMath1 3.1.010. Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x and y variables. Solve the system by the elimination method. Be sure to state your final answer in terms of the original question. The concession stand at an ice hockey rink had receipts of $6200 from selling a total of 2600 sodas and hot dogs. If each soda sold for $2 and each hot dog sold for $3, how many of each were sold? x = sodas y = hot dogs Show My Work (Optional) 8. -/6 pointsBerrFinMath1 3.2.006. Carry out the row operation on the matrix. R1 R2 R1 on 4 9 50 5 7 52 Show My Work (Optional) 9. -/6 pointsBerrFinMath1 3.2.007. Carry out the row operation on the matrix. 4 2 43 1 R2 R2 on 7 0 7 70 Show My Work (Optional) https://www.webassign.net/web/Student/Assignment-Responses/last?dep=13959992 2/3 7/8/2016 Unit 2 Quiz (3.1, 3.2) 10.-/1 pointsBerrFinMath1 3.2.008. Interpret the augmented matrix as the solution of a system of equations. (Enter your answers as a commaseparated list. If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.) 1 0 6 0 1 3 (x, y) = Show My Work (Optional) 11.-/1 pointsBerrFinMath1 3.2.011. Solve the system by rowreducing the corresponding augmented matrix. (Enter your answers as a commaseparated list. If the system is inconsistent, enter INCONSISTENT. If the system is dependent, enter DEPENDENT.) 2x + y = 20 x + y = 14 (x, y) = Show My Work (Optional) 12.-/2 pointsBerrFinMath1 3.2.015. Express the situation as a system of two equations in two variables. Be sure to state clearly the meaning of your x and yvariables. Solve the system by rowreducing the corresponding augmented matrix. State your final answer in terms of the original question. For the final days before the election, the campaign manager has a total of $47,000 to spend on TV and radio campaign advertisements. Each TV ad costs $3000 and is seen by 10,000 voters, while each radio ad costs $500 and is heard by 2000 voters. Ignoring repeated exposures to the same voter, how many TV and radio ads will contact 164,000 voters using the allocated funds? x = TV ads y = radio ads Show My Work (Optional) https://www.webassign.net/web/Student/Assignment-Responses/last?dep=13959992 3/3 Systems of Equations Systems of Equations Graphical Representations of Equations Elimination Method Matrix Representations of Equations Systems of Equations A jar contains a total of 60 coins, comprised entirely of pennies and nickels. The jar contains exactly $2.40. Believe it or not, this is enough information to algebraically determine exactly how many of each coin are in the jar! STEP 1: Write down what you know (# of pennies) + (#of nickels) = 60 (# of pennies)(.01) + (# of nickels)(.05) = $2.40 STEP 2: Now make those into equations LET: p = # of pennies n = # of nickels THIS IS A SYSTEM OF EQUATIONS!!! p + n = 60 Two different equations 0.01p + 0.05n = 2.40 information about the same variables and the same situation representing different Solving Systems by Graphing A system of linear equations can be solved by graphing the lines and finding the coordinates of the point where the lines intersect. y = -x + 4 y-int at 4 Slope = -1 Solve the system by graphing: x y 4 4 x 2 y 6 (1,3) STEP 1: Put the equations in y=mx + b form x y 4 y x 4 4x 2 y 2 2 y 4x 2 y 2x 1 STEP 2: Use the b value to place the y-intercept y = 2x + 1 y-int at 1 Slope = 2 and the m value to trace the slope to find an additional point and draw the line. STEP 3: Determine the coordinates of the point of intersection: (1,3) NOTE: There are three possible situations for a system of two linear equations **1** If the lines cross, like in this example, then there is exactly ONE SOLUTION for the system **2** If the lines are parallel, then they never cross, and there is NO SOLUTION for the system **3** If the lines lie on top of each other, then there are INFINITE SOLUTIONS to the system Solving Systems of Equations By Elimination The trick here is to multiply and then add or subtract. 2 x y 12 4 x 3 y 14 So, what you want to do is find a number that you can multiply every part of one of the equation by and then add or subtract each part to the other equation and have one of the variables cancel out. In this case we will multiply the top equation by 3, and then add it to the bottom equation (looking ahead, we are expecting 3y in the top equation to cancel out with the -3y in the bottom equation.) 3[2x + y = 12] 6x + 3y = 36 Now, add the bottom equation to the NEW top equation and solve for x: 6x + 3y = 36 + 4x - 3y = 14 10x = 50 10 10 x=5 Now plug 5 in for x, in either equation and solve for y. 2(5) + y = 12 10 + y = 12 - 10 - 10 y=2 So, x = 5 and y = 2. By Substitution When using substitution the key is to solve ONE of the equations for ONE of the variables. Both of the equations below CAN be solved for either x or y, however one of them lends itself to the process more than the other. Solve for x: subtract y divide by 2 Solve for x: add 3y divide by 4 Solve for y: subtract 2x 2 x y 12 4 x 3 y 14 Solve for y: subtract 4x divide by -3 Step 1: solve 2x + y = 12 for y y = 12 - 2x Step 2: Now substitute (12 - 2x) into the OTHER equation everywhere you see a y 4x - 3y = 14 4x - 3(12 - 2x) = 14 Step 3: Solve for x 4x - 36 + 6x = 14 10x - 36 = 14 10x = 50 x=5 Step 4: Now plug x = 5 into one of the original equations to solve for y. 2x + y = 12 2(5) + y = 12 y=2 So, x = 5 and y = 2 Sample Problem: Investments A person invests a total of $10,000 in two savings accounts. One account yields 8% simple interest and the other account 10% simple interest. He earns a total of $970 in interest for the year. How much money was invested in the 8% account? Let: The amount invested in the 8% account = x The amount invested in the 10% account = y Equations: x + y = 10,000 0.08x + 0.10y = 970 Using substitution: Step 1: solve the first equation for y y = 10,000 - x Step 2: plug 10,000 - x in for y in the second equation 0.08x + 0.10(10,000 - x) = 970 Step 3: now solve 0.08x + 0.10(10,000 - x) = 970 0.08x + 1,000 - 0.10x = 970 1,000 - 0.02x = 970 30 = 0.02x $1500 = x Sample Problem: The Money Jar A jar contains a total of 60 coins, comprised entirely of pennies and nickels. The jar contains exactly $2.40. Let: The number of pennies = p The number of nickels = n Equations: p + n = 60 0.01p + 0.05n = 2.40 Graphically: NOTE: p is labeled along the x-axis and n is labeled along the y-axis. n Interpreting the solution: (15, 45) (p, n) p There are 15 pennies in the jar and 45 nickels. (15, 45) p n 60 n p 60 0.01 p 0.05n 2.40 0.01 2.40 n p 0.05 0.05 1 n p 48 5 Matrices Position is denoted by the lower case letter of the name of the matrix and the subscript numerals for the row and the column, respectively. 1 2 3 A 4 5 6 7 8 9 a11 1; a21 4; a31 7; a12 2; a22 5; a32 8; a13 3; a23 6; a33 9; MATRICES - The Basics Matrix Dimensions: # of rows x # of columns Example: Augmented Matrices: Represent the following system of equations with an augmented matrix: 3x 5 y 21 2 x 7 y 17 The matrix below has 3 rows and 4 columns It is a 3 x 4 matrix. columns 1 2 3 4 1 rows 2 3 1 2 1 6 5 6 3 2 7 1 4 1 Write the coefficients of the variables, lining the x coefficients up in one column and the y coefficients in their own column, the create a final column with the constants in it. 21 3 5 2 7 17 Matrix Row Operations There are three types of matrix row operations: - Switch any two rows: R1 R2 - Multiply or divide one of the rows by a nonzero number: 3R1 R1 - Replace a row by its sum or difference with a multiple of another row: R1 + R2 R1 Sample Row Operations Switch any two rows Multiply or divide one of the rows by a non zero number Replace a row by its sum or difference with a multiple of another row 5 4 9 R1 R2 3 1 7 2 8 6 3 1 7 5 4 9 2 8 6 9 5 4 9 5 4 4 R3 R3 3 1 7 3 1 7 2 8 6 8 32 24 5 4 9 5 2(3) 4 2(1) 9 2(7) 1 6 23 R1 2 R2 R1 3 1 7 3 1 7 3 1 7 2 8 6 2 8 6 2 8 6 Solving Equations by Row Reduction When solving a system of equations using matrix row reduction, the goal is: \"1s on the diagonal, and 0s above and below the 1s\" This is what it will look like: Notice the 1s on the diagonal! 1 0 18 0 1 7 When this is \"put back into equation form\" it looks like: 1x + 0y = 18 0x + 1y = 7 Which means x = 18 and y = 7 The \"1s\" give us the variable and the answer based on their location! Like coordinates! Sample Problem A high school football concession stand grossed $489 from the sale of sodas and slushies. The sodas cost $1.00 and the slushies cost $1.50. The same size cup was used to serve both, and 441 cups were used. Assuming no cup waste or human error (mischarges, freebies), how many sodas and how many slushies were sold? Define the variables: LET: x = # of sodas sold y = # of slushies sold Write the equations: x + y = 441 1x + 1.5y = 489 Create the augmented matrix: Perform Row Reduction: 1 1 441 1 1.5 489 Step 1 is done! There is already a 1 in the a11 Step 2: we need to add/subt. a row to get a 0 UNDER the 1 from step 1. 1 441 1 1 441 1 1 441 1 R2 R1 R2 1 1.5 489 1 1 1.5 1 489 441 0 0.5 48 Step 3: we need 1s on the diagonal, MULT. or DIV. to change the 0.5 1 441 1 1 441 1 1 441 1 2 R2 R2 0 0.5 48 2(0) 2(0.5) 2(48) 0 1 96 Step 4: we need to add/subt. a row to get a 0 ABOVE the 1 from step 3. 1 1 441 1 0 1 1 441 96 1 0 345 R1 R2 R1 1 96 0 1 96 0 1 96 0 1 1 441 1 1.5 489 Step 5: Use the 1s to determine the answer 1x 0 y 345 SO 0 x 1 y 96 x 345 sodas y 96 slushies You can now... Solve systems of equations using: - Algebra - Graphs - Matrices Unit 2 Systems of Equations and Matrices - Sample Problems 3.1 Systems of Two Linear Equations in Two Variables 1. Solve the system by graphing: y 3x 15 y 2 x 10 The solution to the system is the point where the lines cross y = 3x - 15 y -int: -15 m: 3 y = - 2x + 10 y-int: 10 m: -2 The graph shows the two lines, they cross at (5,0) 40 30 20 10 -10 -5 0 0 -10 -20 -30 -40 2. Solve the system by the elimination method. 5 10 15 20 y=3x-15 y=-2x+10 x 3 y 16 2x y 7 Row 2 + 2 Row 1 x 3 y 16 5 y 25 Divide Row 2 by 5 x 3 y 16 y5 Row 1 + 3 Row 2 x 1 y 5 Test x 3 y 16 1 3(5) 16check ! 2x y 7 2(1) 5 7check ! 3. Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x- and y- variables. Solve the system by the elimination method (this is not the only method - we'll show all three!). Be sure to state your final answer in terms of the original question. A movie theater sold 1500 tickets to its opening day showings of Thor. The student ticket price was $7.00 and the adult ticket price was $9.50. The total ticket revenue for the day was $12,517.50. How many student tickets and how many adult tickets were sold? Define variables: x = # of student tickets y = # of adult tickets Write linear equations: The total number of tickets sold is 1500, so: The total revenue is $12,517.50, so: Solve the system by elimination: x y 1,500 7 x 9.5 y 12,517.5 Row 2 7 Row 1 x y 1500 2.5 y 2017.5 Divide Row 2 by 2.5 x y 1500 y 807 Row 1 Row 2 x 693 y 807 OR Solve the system by substitution: x + y = 1500 $7.00x + $9.50y = $12,517.50 x y 1,500 7 x 9.5 y 12,517.5 x y 1500 x 1500 y Substitute 1500 y in for x in the second equation: 7 x 9.5 y 12,517.50 7(1500 y ) 9.5 y 12,517.5 Now solve for x : 7(1500 y ) 9.5 y 12,517.5 10,500 7 y 9.5 y 12,517.5 10,500 2.5 y 12,517.5 2.5 y 2017.5 y 807 Now substitute 807 in for y in the first equation: x y 1500 x 807 1500 x 693 OR Solve the system graphically (with some help from Excel!!): x + y = 1500 $7.00x + $9.50y = $12,517.50 The final answer, no matter how you solved it: 693 students and 807 adults bought tickets to the opening day shows of Thor (at this theater)! 3.2 Matrices and Linear Equations in Two Variables 1. Find the dimensions of the matrix and the values of a1,1 and a2,3 8 5 9 6 2 7 Dimensions = row x column = 2 x 3; a1,1 = -8; a2,3 = 7 2. Write the augmented matrix representing the system of linear equations. 2 x 3y 4 3 y 5 2 3 4 Augmented Matrix: 3 0 1 5 3. Carry out the row operation: 1 3 4 1 R2 R2 2 0 2 6 1 3 4 1 3 4 1 R2 R2 2 0 2 6 0 1 3 4. Interpret the augmented matrix as the solution of a system of equations. State the solution, or identify the system as inconsistent or dependent. 1 0 4 0 1 5 Solution: x = 4 and y = -5 5. Solve the system by row-reducing the corresponding augmented matrix. x 2y 5 3x y 1 1 2 5 3 1 1 R2 3R1 1 1 R2 7 0 5 0 7 14 1 2 2 5 1 2 1 0 1 R1 2 R2 0 1 2 x 1 y2 6. A street vendor has a total of 350 short and long sleeve T-shirts. If she sells the short sleeve shirts for $10 each and the long sleeve shirts for $14 each, how many of each did she sell if she sold all of her stock for $4300? Let: x = the number of short sleeve shirts y = the number of long sleeve shirts x y 350 10 x 14 y 4300 1 1 350 10 14 4300 1 1 350 R2 14 R1 4 0 600 R2 1 1 350 4 1 0 150 0 1 200 R1 R2 1 0 150 x 150 y 200 Systems of Equations Systems of Equations Graphical Representations of Equations Elimination Method Matrix Representations of Equations Systems of Equations A jar contains a total of 60 coins, comprised entirely of pennies and nickels. The jar contains exactly $2.40. Believe it or not, this is enough information to algebraically determine exactly how many of each coin are in the jar! STEP 1: Write down what you know (# of pennies) + (#of nickels) = 60 (# of pennies)(.01) + (# of nickels)(.05) = $2.40 STEP 2: Now make those into equations LET: p = # of pennies n = # of nickels THIS IS A SYSTEM OF EQUATIONS!!! p + n = 60 Two different equations 0.01p + 0.05n = 2.40 information about the same variables and the same situation representing different Solving Systems by Graphing A system of linear equations can be solved by graphing the lines and finding the coordinates of the point where the lines intersect. y = -x + 4 y-int at 4 Slope = -1 Solve the system by graphing: x y 4 4 x 2 y 6 (1,3) STEP 1: Put the equations in y=mx + b form x y 4 y x 4 4x 2 y 2 2 y 4x 2 y 2x 1 STEP 2: Use the b value to place the y-intercept y = 2x + 1 y-int at 1 Slope = 2 and the m value to trace the slope to find an additional point and draw the line. STEP 3: Determine the coordinates of the point of intersection: (1,3) NOTE: There are three possible situations for a system of two linear equations **1** If the lines cross, like in this example, then there is exactly ONE SOLUTION for the system **2** If the lines are parallel, then they never cross, and there is NO SOLUTION for the system **3** If the lines lie on top of each other, then there are INFINITE SOLUTIONS to the system Solving Systems of Equations By Elimination The trick here is to multiply and then add or subtract. 2 x y 12 4 x 3 y 14 So, what you want to do is find a number that you can multiply every part of one of the equation by and then add or subtract each part to the other equation and have one of the variables cancel out. In this case we will multiply the top equation by 3, and then add it to the bottom equation (looking ahead, we are expecting 3y in the top equation to cancel out with the -3y in the bottom equation.) 3[2x + y = 12] 6x + 3y = 36 Now, add the bottom equation to the NEW top equation and solve for x: 6x + 3y = 36 + 4x - 3y = 14 10x = 50 10 10 x=5 Now plug 5 in for x, in either equation and solve for y. 2(5) + y = 12 10 + y = 12 - 10 - 10 y=2 So, x = 5 and y = 2. By Substitution When using substitution the key is to solve ONE of the equations for ONE of the variables. Both of the equations below CAN be solved for either x or y, however one of them lends itself to the process more than the other. Solve for x: subtract y divide by 2 Solve for x: add 3y divide by 4 Solve for y: subtract 2x 2 x y 12 4 x 3 y 14 Solve for y: subtract 4x divide by -3 Step 1: solve 2x + y = 12 for y y = 12 - 2x Step 2: Now substitute (12 - 2x) into the OTHER equation everywhere you see a y 4x - 3y = 14 4x - 3(12 - 2x) = 14 Step 3: Solve for x 4x - 36 + 6x = 14 10x - 36 = 14 10x = 50 x=5 Step 4: Now plug x = 5 into one of the original equations to solve for y. 2x + y = 12 2(5) + y = 12 y=2 So, x = 5 and y = 2 Sample Problem: Investments A person invests a total of $10,000 in two savings accounts. One account yields 8% simple interest and the other account 10% simple interest. He earns a total of $970 in interest for the year. How much money was invested in the 8% account? Let: The amount invested in the 8% account = x The amount invested in the 10% account = y Equations: x + y = 10,000 0.08x + 0.10y = 970 Using substitution: Step 1: solve the first equation for y y = 10,000 - x Step 2: plug 10,000 - x in for y in the second equation 0.08x + 0.10(10,000 - x) = 970 Step 3: now solve 0.08x + 0.10(10,000 - x) = 970 0.08x + 1,000 - 0.10x = 970 1,000 - 0.02x = 970 30 = 0.02x $1500 = x Sample Problem: The Money Jar A jar contains a total of 60 coins, comprised entirely of pennies and nickels. The jar contains exactly $2.40. Let: The number of pennies = p The number of nickels = n Equations: p + n = 60 0.01p + 0.05n = 2.40 Graphically: NOTE: p is labeled along the x-axis and n is labeled along the y-axis. n Interpreting the solution: (15, 45) (p, n) p There are 15 pennies in the jar and 45 nickels. (15, 45) p n 60 n p 60 0.01 p 0.05n 2.40 0.01 2.40 n p 0.05 0.05 1 n p 48 5 Matrices Position is denoted by the lower case letter of the name of the matrix and the subscript numerals for the row and the column, respectively. 1 2 3 A 4 5 6 7 8 9 a11 1; a21 4; a31 7; a12 2; a22 5; a32 8; a13 3; a23 6; a33 9; MATRICES - The Basics Matrix Dimensions: # of rows x # of columns Example: Augmented Matrices: Represent the following system of equations with an augmented matrix: 3x 5 y 21 2 x 7 y 17 The matrix below has 3 rows and 4 columns It is a 3 x 4 matrix. columns 1 2 3 4 1 rows 2 3 1 2 1 6 5 6 3 2 7 1 4 1 Write the coefficients of the variables, lining the x coefficients up in one column and the y coefficients in their own column, the create a final column with the constants in it. 21 3 5 2 7 17 Matrix Row Operations There are three types of matrix row operations: - Switch any two rows: R1 R2 - Multiply or divide one of the rows by a nonzero number: 3R1 R1 - Replace a row by its sum or difference with a multiple of another row: R1 + R2 R1 Sample Row Operations Switch any two rows Multiply or divide one of the rows by a non zero number Replace a row by its sum or difference with a multiple of another row 5 4 9 R1 R2 3 1 7 2 8 6 3 1 7 5 4 9 2 8 6 9 5 4 9 5 4 4 R3 R3 3 1 7 3 1 7 2 8 6 8 32 24 5 4 9 5 2(3) 4 2(1) 9 2(7) 1 6 23 R1 2 R2 R1 3 1 7 3 1 7 3 1 7 2 8 6 2 8 6 2 8 6 Solving Equations by Row Reduction When solving a system of equations using matrix row reduction, the goal is: \"1s on the diagonal, and 0s above and below the 1s\" This is what it will look like: Notice the 1s on the diagonal! 1 0 18 0 1 7 When this is \"put back into equation form\" it looks like: 1x + 0y = 18 0x + 1y = 7 Which means x = 18 and y = 7 The \"1s\" give us the variable and the answer based on their location! Like coordinates! Sample Problem A high school football concession stand grossed $489 from the sale of sodas and slushies. The sodas cost $1.00 and the slushies cost $1.50. The same size cup was used to serve both, and 441 cups were used. Assuming no cup waste or human error (mischarges, freebies), how many sodas and how many slushies were sold? Define the variables: LET: x = # of sodas sold y = # of slushies sold Write the equations: x + y = 441 1x + 1.5y = 489 Create the augmented matrix: Perform Row Reduction: 1 1 441 1 1.5 489 Step 1 is done! There is already a 1 in the a11 Step 2: we need to add/subt. a row to get a 0 UNDER the 1 from step 1. 1 441 1 1 441 1 1 441 1 R2 R1 R2 1 1.5 489 1 1 1.5 1 489 441 0 0.5 48 Step 3: we need 1s on the diagonal, MULT. or DIV. to change the 0.5 1 441 1 1 441 1 1 441 1 2 R2 R2 0 0.5 48 2(0) 2(0.5) 2(48) 0 1 96 Step 4: we need to add/subt. a row to get a 0 ABOVE the 1 from step 3. 1 1 441 1 0 1 1 441 96 1 0 345 R1 R2 R1 1 96 0 1 96 0 1 96 0 1 1 441 1 1.5 489 Step 5: Use the 1s to determine the answer 1x 0 y 345 SO 0 x 1 y 96 x 345 sodas y 96 slushies You can now... Solve systems of equations using: - Algebra - Graphs - Matrices Unit 2 Systems of Equations and Matrices - Sample Problems 3.1 Systems of Two Linear Equations in Two Variables 1. Solve the system by graphing: y 3x 15 y 2 x 10 The solution to the system is the point where the lines cross y = 3x - 15 y -int: -15 m: 3 y = - 2x + 10 y-int: 10 m: -2 The graph shows the two lines, they cross at (5,0) 40 30 20 10 -10 -5 0 0 -10 -20 -30 -40 2. Solve the system by the elimination method. 5 10 15 20 y=3x-15 y=-2x+10 x 3 y 16 2x y 7 Row 2 + 2 Row 1 x 3 y 16 5 y 25 Divide Row 2 by 5 x 3 y 16 y5 Row 1 + 3 Row 2 x 1 y 5 Test x 3 y 16 1 3(5) 16check ! 2x y 7 2(1) 5 7check ! 3. Formulate the situation as a system of two linear equations in two variables. Be sure to state clearly the meaning of your x- and y- variables. Solve the system by the elimination method (this is not the only method - we'll show all three!). Be sure to state your final answer in terms of the original question. A movie theater sold 1500 tickets to its opening day showings of Thor. The student ticket price was $7.00 and the adult ticket price was $9.50. The total ticket revenue for the day was $12,517.50. How many student tickets and how many adult tickets were sold? Define variables: x = # of student tickets y = # of adult tickets Write linear equations: The total number of tickets sold is 1500, so: The total revenue is $12,517.50, so: Solve the system by elimination: x y 1,500 7 x 9.5 y 12,517.5 Row 2 7 Row 1 x y 1500 2.5 y 2017.5 Divide Row 2 by 2.5 x y 1500 y 807 Row 1 Row 2 x 693 y 807 OR Solve the system by substitution: x + y = 1500 $7.00x + $9.50y = $12,517.50 x y 1,500 7 x 9.5 y 12,517.5 x y 1500 x 1500 y Substitute 1500 y in for x in the second equation: 7 x 9.5 y 12,517.50 7(1500 y ) 9.5 y 12,517.5 Now solve for x : 7(1500 y ) 9.5 y 12,517.5 10,500 7 y 9.5 y 12,517.5 10,500 2.5 y 12,517.5 2.5 y 2017.5 y 807 Now substitute 807 in for y in the first equation: x y 1500 x 807 1500 x 693 OR Solve the system graphically (with some help from Excel!!): x + y = 1500 $7.00x + $9.50y = $12,517.50 The final answer, no matter how you solved it: 693 students and 807 adults bought tickets to the opening day shows of Thor (at this theater)! 3.2 Matrices and Linear Equations in Two Variables 1. Find the dimensions of the matrix and the values of a1,1 and a2,3 8 5 9 6 2 7 Dimensions = row x column = 2 x 3; a1,1 = -8; a2,3 = 7 2. Write the augmented matrix representing the system of linear equations. 2 x 3y 4 3 y 5 2 3 4 Augmented Matrix: 3 0 1 5 3. Carry out the row operation: 1 3 4 1 R2 R2 2 0 2 6 1 3 4 1 3 4 1 R2 R2 2 0 2 6 0 1 3 4. Interpret the augmented matrix as the solution of a system of equations. State the solution, or identify the system as inconsistent or dependent. 1 0 4 0 1 5 Solution: x = 4 and y = -5 5. Solve the system by row-reducing the corresponding augmented matrix. x 2y 5 3x y 1 1 2 5 3 1 1 R2 3R1 1 1 R2 7 0 5 0 7 14 1 2 2 5 1 2 1 0 1 R1 2 R2 0 1 2 x 1 y2 6. A street vendor has a total of 350 short and long sleeve T-shirts. If she sells the short sleeve shirts for $10 each and the long sleeve shirts for $14 each, how many of each did she sell if she sold all of her stock for $4300? Let: x = the number of short sleeve shirts y = the number of long sleeve shirts x y 350 10 x 14 y 4300 1 1 350 10 14 4300 1 1 350 R2 14 R1 4 0 600 R2 1 1 350 4 1 0 150 0 1 200 R1 R2 1 0 150 x 150 y 200