G SYSTEMS OF EQUATIONS WORKSHEET 1 - GRAPHING AND SUBSTITUTION

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1 1 Name: GRADE 11 PRE-CALCULUS Date: UNIT G SYSTEMS OF EQUATIONS WORKSHEET 1 - GRAPHING AND SUBSTITUTION 1. Solve the following system of equations by graphing. y = x - and y = -x + 10 both equations by evaluation Eqn1: 3=(3.5) -? YES Eqn 3=-(3.5) + 10?YES 3.5, 3 Ans: (3.5, 3). Solve the following system of equations by graphing y = x y = 3x + both equations Eqn1: - = -?? YES Eqn: -=3(-) +?? YES -, - Ans: (-, -) Gr11Pre_G_SysLinEq_Worksheet_1_KEY.doc

2 3. Solve the following system of equations by graphing. x + y = - -x + y = 10 both equations -3, x 0 - Line 1 y - 0 Line x y Ans: (-3, ). Solve the following systems of equations using the substitution method. Don t forget to check your solution in both equations. a. y = x and y = -x + 10 Four possible ways to do it! Yours may be different, but will get the same answer. The easiest way is this: (Eqn1) y = x and (Eqn) y = -x + 10 x = -x + 10 Now we have one unknown and one equation! Now just use Algebra! x + x = 10 + x = 1 x 1 = x = 3.5 Substitute the now known x back into either Eqn 1 or Eqn to find y by evaluating (Eqn1) y = (3.5) - = 7 = 3 Solution: (3.5, 3)

3 3 b. y = x and y = 3x + Eqn 1: y = x and Eqn : y = 3x + There are several ways to do this. The easiest is just to say: if y = x and y also equals 3x +, then x = 3x + Therefore x - 3x = So -x =. So x = - Substitute the now known x back into either equation 1 or equation to find the other unknown. y=x = -. y = - Solution: (-, -) c. x + y = - and -x + y = 10 There are again possible ways to do this. Just isolate one variable though by itself in one equation so you can substitute it into the other equation EQN1: x + y = - EQN: -x + y = 10 Isolate the y in Eqn 1 seems easy by subtracting x from both sides: Eqn1: y = - - x Substitute that y into the other equation; Eqn -x + (- - x) = 10 Now we have only one unknown! And one equation! -x = 10 Therefore -x=1. x = -3 Sticking the x back into either eqn we find that: (-3) + y = - so that y= Solution: -3,

4 5. There are two cab companies that work at the airport. Jackie s Limos and Walter s Luxury Cabs. Jackie s charges $3.50 flat rate (as soon as you sit in the seat) and 0 cents per kilometre (Km). Walter s charges $1.50 flat rate and 0 cents per Km. a. write the linear equations for both taxi services. (Cost as a function of distance). Cost Jackie = x Cost Walter = x Cost Your graph may look a bit different depending on what scale you used b. plot the graph that represents both taxi services. Label each line 10 Distance c. From the graph; at what distance do the two companies charge the same amount? At 10Km they both charge $7.50 d. Using the substitution method find at what distance do the two companies charge the same amount? x = x = 0.x x=10 if x = 10 then substituting back to evaluate in either eqn: y = (10)= 7.50 Solution (10km, $7.50) e. If you were traveling only 5 Km which taxi would you take? How much does each company charge for a 5 Km ride? Walter is cheaper for 5km, (but Cost Jackie = x = (5) = $5.50 after 10 km, Jackie is cheaper) Cost Walter = (5) = $.50 Ans: c;d:10km. e. Walter s, $5.50, $.50

5 5. Two numbers, a and b, add to give and have a difference of 30. Find the two numbers using the substitution method. a + b = therefore a = - b a - b=30 ( b) b = 30 -b=30 therefore 1=b. So b = 7 Therefore a must equal 37 found by substituting b=7 back into either equation Solution: 37 and 7 Ans: 37 and 7 7. Ray gets his morning exercise by canoeing. Every morning he paddles upstream 1.5 km in 30 minutes and then paddles down-stream in 1 minutes. What is the speed of the current in Km/hour and what speed does Ray paddle in Km/hour? We want to know speed. Speed upstream is your paddle speed P subtract the current speed C. Speed going downstream is paddle speed P + current speed C. The speed going up stream is 3 Km/hr (since 1.5/0.5 = 3) and the speed going down stream is 1.5/0.3 or 5 km/hr. 3=P-C so P = 3 + C 5 = P + C therefore: 5 = (3 + C) + C so 5=3+C or = C. C=1 (the current = 1 Km/Hr) So then Ray paddles at KM/Hr! Solution: Ray paddles at Km/Hr, the current is 1 Km/hr Ans: Ray paddles at km/hour, the current is 1 Km / hr

Topic 5: Statistics 5.3 Cumulative Frequency Paper 1

Topic 5: Statistics 5.3 Cumulative Frequency Paper 1 1. The following is a cumulative frequency diagram for the time t, in minutes, taken by students to complete a task. Standard Level Write down the median.