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Mastery Learning Algebra 1 Systems of Equations, Direct Variation, Square Root Systems of Equations Solve by matrix: Write both equations with X first then Y (on the same side) You Try 1. Given 2x 3y 12 6x 2y 42 What is x+y? Ax+By=C CALCULATOR: [2nd] [MATRIX]->2x3->Enter coefficients of all six[2nd][QUIT]->[2nd][MATRIX]>MATH->B:rref(->]->[2nd][MATRIX]A[ENTER] Types of Solutions: 1 0 # x (# x ,# y ) 0 1 # y 1 0 # NO SOLUTION: 0 01 1 0 # x MANY SOLUTIONS: 0 0 0 ONE SOLUTION: If a variable is missing then put a ‘0’ If the number in front of a variable is missing then put a ‘1; Ex: x = 3y - 2 & 4x = 16 1x – 3y = -2 4x + 0y = 16 1 3 2 and then do RREF[A] 16 4 0 1 0 4 and you get so x = 4 and y = 2 0 1 2 So [A] is Solve by Graphing: Solve for y, graph find intersection. Direct Variation y=kx K is the constant of variation, plug in values for x and y to solve for k. Then use second value for x or y to solve for missing variable. EX: If y varies directly as x, and y=15 when x=-60, then what is the value of y when x=256? Use y=kx 15=-60k restaurant received 550 hamburgers and 425 hotdogs for $630. a. How much did each hamburger cost? b. How much will 25 hamburgers and 50 hotdogs be? 3. A local pet store has triple the amount of fish as birds and has a total of 250 fish and birds. Write a system of equations that represents the number of fish and birds using the variables F and B. 4. Given the system of equations 4x 3y 60 x y 10 what is the value of x? 5. Given: A. 11 2x + y = 15 5x - 6y = -22 What is the value of x - y? B. 2 C. 3 D. -3 6. Given: w = 1 - v 2v + w = 4 Find the value of w. A. 3 B. 2 C. 1 D. -2 7. A limousine company charges a flat-fee of $80 plus $.05 per mile. A shuttle van company charges a flat-fee of $60 plus $.50 per mile. Approximately what mileage will yield the same fare for both? A. 24 miles B. 34 miles C. 44 miles D. 54 miles 8. If y varies directly as x, and y=20 when x=-2, then what is the value of y when x=15? 9. If y varies directly as x, and y = 10 when x = 4, then what is the value of x when y = 25. 10. The number of gallons of gasoline varies directly with the cost per gallon. It costs $75 for 20 gallons 1 of gasoline. How much does it cost for 13 gallons? k= So…substitute the value of k into y=kx 4 with given value of x 2. A restaurant received 270 hamburger patties and 350 hotdogs on Monday for $450. On Friday the 1 4 y=256( ) y=-64 11. A factory can produce 123 radios in 2 weeks. Assuming direct variation how many radios can be produced in 5 weeks? Mastery Learning Algebra 1 Systems of Equations, Direct Variation, Square Root Square Root Simplifying Radical Expressions: We don’t like decimals all the time. Find the prime factorization of the radicand SQUARE ROOTS are like a house with party on the inside and only couples (pairs) can leave together. o All pairs of like numbers or variables are multiplied in front of the square root. o All the single numbers or variables remain inside the square root. o If no singles remain, then don’t write square root. The radicand is called a perfect square. EX: 136 Prime Factorization of 136= 2 2 2 17 . There is one pair of 2’s ANSWER: 2 2 17 2 34 12. The length of a spring varies directly as the weight attached. If a spring stretches 1.6 inches when a 24 pound weight is attached, how far will the spring stretch when a 15 pound weight is attached? A. 3.9 inches B. 2.56 inches C. 2.25 inches D. 1 inch 13. Police officers can use the formula s 30 fd to determine the speed s that a car was traveling in miles per hour by measuring the distance d in feet of its skid marks. In this formula, f is the coefficient of friction for the type and condition of the road. a. Write a simplified expression for the speed if f = 0.6 for a wet asphalt roaD. b. What is a simplified expression for the speed if f = 0.8 for a dry asphalt road? c. An officer measures skid marks that are 110 feet long. Determine the speed of the car for both wet road conditions and for dry road conditions. 14. Find the area of the rectangle if l=( 3 ) and w=( 12 5 ). A. EX: 5 C. 6 3 3 5 2 72 x y 72 2 2 2 3 3 x5 x x x x x y2 y y Calculator method for multiple choice: Type in the original problem and write down the decimal. Then check all the answer choices to see which one also gives the same decimal answer as the problem. The Distance formula d ( x 2 x1 ) 2 ( y 2 y1 ) 2 (( x2 x1 ) 2 ( y2 y1 ) 2 B. 6 + D. 15 51 15. The side of a square is connected between (1,-2) and (-3, 1). Find the square’s perimeter. (Use the distance formula to find one side) A. A. 20 units B. 25 units ANSWER 6 x 2 y 2 x Calculator: d 36 15 C. 4 7 units D. 7 units. 16. Voltage V is given by V PR where P is the power in Watts and R is the resistance in ohms. How many more volts are needed to light a 75-watt bulb than a 50-watt bulb if the resistance for both is 100 ohms? (Hint: Which letters of the equation can substitute for numbers?) 17. Simplify: A. 2 54 x 4 y 7 6 x 2 y 3 6 y B. 2 x 2 y 3 6 y C. 2 x 2 6y7 D. 2