University:
Shippensburg University of Pennsylvania
Course:
MAT 211 | Calculus I
Academic year:

2024
Views:

403

Pages:

5
Author:

Herbosambx

Question 1 Solve the system of equations. 𝑦 = 2 𝑥 − 5 y=2x−5 3 𝑥 + 2 𝑦 = 18 3x+2y=18 Answer Step 1 Given system of equations: 𝑦 = 2 𝑥 − 5 ( 1 ) y=2x−5(1) 3 𝑥 + 2 𝑦 = 18 ( 2 ) 3x+2y=18(2) Step 2 Substituting 𝑦 y from (1) into (2): 3 𝑥 + 2 ( 2 𝑥 − 5 ) = 18 3x+2(2x−5)=18 3 𝑥 + 4 𝑥 − 10 = 18 3x+4x−10=18 7 𝑥 = 28 7x=28 𝑥 = 4 x=4 Substituting 𝑥 = 4 x=4 in (1): 𝑦 = 2 ( 4 ) − 5 = 8 − 5 = 3 y=2(4)−5=8−5=3 Step 3 Thus, the solution to the given system of equations is ( 4 , 3 ) (4,3). Question 2 Consider the following system of linear equations: 4 𝑥 + 2 𝑦 = 25 4x+2y=25 4 𝑥 − 𝑦 = − 5 4x−y=−5 If the value of 𝑦 y is 10 10, what is the value of 𝑥 x for this system? Options: 1.25 1.25 11.25 11.25 1.45 1.45 4.5 4.5 Answer Step 1 Given the system of equations: 4 𝑥 + 2 𝑦 = 25 4x+2y=25 4 𝑥 − 𝑦 = − 5 4x−y=−5 If 𝑦 = 10 y=10, then substitute into the first equation: 4 𝑥 + 2 ( 10 ) = 25 4x+2(10)=25 4 𝑥 + 20 = 25 4x+20=25 4 𝑥 = 5 4x=5 𝑥 = 5 4 = 1.25 x= 4 5 =1.25 Answer: 𝑥 = 1.25 x=1.25. Question 3 Solve for 𝑥 x and 𝑦 y: 3 𝑥 + 12 𝑦 = − 30 ( 1 ) 3x+12y=−30(1) 6 𝑥 + 6 𝑦 = − 24 ( 2 ) 6x+6y=−24(2) Answer 1 Step 1 Using the method of addition: Multiply (1) by 2 2: 2 ( 3 𝑥 + 12 𝑦 ) = 6 𝑥 + 24 𝑦 = − 60 ( 3 ) 2(3x+12y)=6x+24y=−60(3) Now, subtract equation (2) from equation (3): 6 𝑥 + 24 𝑦 − ( 6 𝑥 + 6 𝑦 ) = − 60 − ( − 24 ) 6x+24y−(6x+6y)=−60−(−24) 18 𝑦 = − 36 18y=−36 𝑦 = − 2 y=−2 Step 2 Substitute 𝑦 = − 2 y=−2 into (2): 6 𝑥 + 6 ( − 2 ) = − 24 6x+6(−2)=−24 6 𝑥 − 12 = − 24 6x−12=−24 6 𝑥 = − 12 6x=−12 𝑥 = − 2 x=−2 Solution: 𝑥 = − 2 , 𝑦 = − 2 x=−2,y=−2 Question 4 Compute 4.659 × 1 0 4 − 2.14 × 1 0 4 4.659×10 4 −2.14×10 4 . Round the answer appropriately. Express your answer as an integer using the proper number of significant figures. Answer 1 4.659 × 1 0 4 − 2.14 × 1 0 4 4.659×10 4 −2.14×10 4 = ( 4.659 − 2.14 ) × 1 0 4 =(4.659−2.14)×10 4 = 2.519 × 1 0 4 =2.519×10 4 After the decimal point, there are two significant digits in 2.14 2.14. Thus, the final answer should have two significant figures: Answer: 2.52 × 1 0 4 or 25 , 200. Answer: 2.52×10 4 or 25,200. Question 5 Change the polar equation 5 𝑟 = sin ⁡ ( 𝜃 ) 5r=sin(θ) to Cartesian equation (an equation that only contains 𝑥 x and 𝑦 y). Answer Step 1 Given polar equation: 5 𝑟 = sin ⁡ ( 𝜃 ) 5r=sin(θ) We know: 𝑥 = 𝑟 cos ⁡ ( 𝜃 ) , 𝑦 = 𝑟 sin ⁡ ( 𝜃 ) , and 𝑥 2 + 𝑦 2 = 𝑟 2 x=rcos(θ),y=rsin(θ),and x 2 +y 2 =r 2 Step 2 Multiply 𝑟 r on both sides: 5 𝑟 2 = 𝑟 sin ⁡ ( 𝜃 ) 5r 2 =rsin(θ) Substitute 𝑟 2 = 𝑥 2 + 𝑦 2 r 2 =x 2 +y 2 and 𝑟 sin ⁡ ( 𝜃 ) = 𝑦 rsin(θ)=y: 5 ( 𝑥 2 + 𝑦 2 ) = 𝑦 5(x 2 +y 2 )=y Step 3 Rearrange: 5 𝑥 2 + 5 𝑦 2 − 𝑦 = 0 5x 2 +5y 2 −y=0 5 𝑥 2 + 𝑦 ( 5 𝑦 − 1 ) = 0 5x 2 +y(5y−1)=0 Final Answer: 5 𝑥 2 + 𝑦 ( 5 𝑦 − 1 ) = 0 5x 2 +y(5y−1)=0