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Math 124 Unit 2 Homework 1. Solve the following systems of equations by the graphing methods. Indicate at least two points used for each line. ( 8 pts) a. 5x – 3y = 9 x + 2y = 7 b. 3x - y = 4 6x – 2y = 12 Points used Line 1 ______________ Points used Line 1 ______________ Points used Line 2 ______________ Points used Line 2 ______________ Solution _____________ Solution __________________ 2. Solve the following system of equations using substitution. If there is a specific solution, state that solution. Otherwise, state no solution or many solutions. (2pts) a. – x + 3y = 2 2x – 6y = -4 b. x + 4y = -1 2x – 5y = 11 3. Solve the following system of equations using the elimination method. If there is a specific solution, state that solution. Otherwise, state no solution or many solutions. (2pts) a. 6x + 3y = 9 -8x -4y = -12 b. x – 3y = -7 3x + 2y = 23 4. A supplier for the electronic industry manufactures keyboards and screens for graphing calculators at plants in Mexico and Taiwan. The hourly production rates at each plant are provided in the following table. How many hours should each plant be operated to exactly fill an order for 4000 keyboards and 4000 screens. (2pts) Plant Mexico Taiwan Keyboards 40 20 Screens 32 32 a. Write a system of equations to represent the given problem. b. Solve the system of equations using either the substitution or the elimination method. 5. A bike shop has fixed costs of $5200 to operate their business. Each bike they sell costs $145 to produce. Their bikes sell for $225 each. ( 3pts) a. Write the cost equation. b. Write the revenue equation. c. How many bikes do they have to sell to break even? 6. Suppose that the Supply and Demand for a product is given by the following equations p = 1.5q + 4.2 p = −2.5q + 15 Supply Equation Demand Equation where p is the price in dollars and q is the quantity in thousands. ( 6 pts) a) Find the supply and demand (to the nearest unit) if the product is priced at $6. Supply __________ Demand ______________ b) Find the supply and demand (to the nearest unit) if the product is priced at $10. Supply _____________ Demand ________________ c) Find the equilibrium price and quantity. Price ______________ Quantity ___________________ 7. State the dimensions of the following matrices. (3pts) a. 3 [4 ] 3 b. [ 1 3 4 ] 3 c. [4 1 2 5 5 6] 2 3 8. Write an augmented matrix to represent the following system of equations. (2pts) a. -3x - 6y + 9z = 12 -x + 2y= -10 x – 8z = -2 b. 4(x -3y) = 9y + 6 8x = 9y + 6 9. Write the system of equations represented by the following augmented matrix. (2pts) 1 0 0 3 a. 0 1 0 0.05 0 0 1 12 8 7 113 b. 2 0 1 3 10. Perform the following row operations on the given matrix. (2pts) 12 0 6 3 a. 2 8 4 18 –R3 + R2 R2 2 1 7 0 2 4 10 b. 5 3 1 1 2 R1R1 11. Write as an augmented matrix and then use row reduction to solve. Show all steps in the row reduction and label each. ( 8 pts) a. 2x + 4y = -7 x – 3y= 9 c. x + 3y – 2z = 5 4x – y + 3z = 7 2x – 7y + 7z = 4 b. x – 6y = 4 5x – 30y = 20 d. 2x – 4y + 12z = 20 -x + 3y + 5z = 15 3x – 7y + 7z = 5 12. Find a,b,c,d for the following. (1pt) [ 4 𝑏 ] + [ 8 𝑑 𝑎 𝑐 a ___________ −8 7 ] = [ −3 1 3 ] 2 b ______________ c _____________ d ______________ 13. Multiply the following matrices. Show all work. (2pts) a. [ 𝑥 −𝑎 𝑦 𝑎 ]x[ 𝑥 −𝑏 𝑏 ] 𝑦 b. [ 𝑦 −𝑏 𝑏 𝑥 ] x[ −𝑎 𝑦 𝑎 ] 𝑥 14. Determine the product of each matrix multiplication and then, state whether the following pairs of matrices are inverses of each other. Calculator may be used for multiplication. (4 pts) 10 −3 1 a. [ ] [ 3 −1 3 3 ] −10 11 2 −8 3 b. [ 4 1 −3] [0 −8 −1 6 4 −4 2 −5 2 1] 3 15. Find the inverse, if possible. Use the method discussed in class, not the formula using the determinant. Each row operation should be clearly marked. ( 4pts) a. [ 1 2 ] 3 5 1 3 b. b. [0 1 3 2 9 4] 3 16. Write each system as a matrix equation and solve by using the inverse matrix method. Show all work for finding the inverse. This includes clearly marking each row operation. A calculator can be used for the multiplication part. (4 pts) a. 1x + 2y – 1z = 2 1x + 1y + 2z = 0 1x – 1y – 1z = 1 b. 1x + 3y =5 2x + 1y = 10