University:
Jackson College
Course:
MAT-010 | Basic Mathematics
Academic year:

2024
Views:

428

Pages:

3
Author:

Bennett Wheeler

Question 1 The given system of equations is: 𝑥 − 4 𝑦 = 8 x−4y=8 𝑦 = 1 4 𝑥 + 3 y= 4 1 x+3 Answer 1 The graph shows that the two lines are parallel, which means they have the same slope and will never intersect. Thus, the system doesn't have any solution. Question 2 Theo buys a slice of pizza and a bottle of water for $3. Ralph buys 3 slices of pizza and 2 bottles of water for $8. How much does a slice of pizza cost? Answer 1 Let 𝑥 x be the cost of a slice of pizza and 𝑦 y be the cost of a bottle of water. Theo's purchase equation: 𝑥 + 𝑦 = 3 x+y=3 Ralph's purchase equation: 3 𝑥 + 2 𝑦 = 8 3x+2y=8 Solve the system: Multiply the first equation by 2: 2 𝑥 + 2 𝑦 = 6 2x+2y=6 Subtract from the second equation: ( 3 𝑥 + 2 𝑦 ) − ( 2 𝑥 + 2 𝑦 ) = 8 − 6 (3x+2y)−(2x+2y)=8−6 𝑥 = 2 x=2 Substitute 𝑥 = 2 x=2 into 𝑥 + 𝑦 = 3 x+y=3: 2 + 𝑦 = 3 2+y=3 𝑦 = 1 y=1 So, the cost of a slice of pizza is $2, and the cost of a bottle of water is $1. Question 3 Solve the system: 3 𝑥 − 𝑦 = 7 3x−y=7 5 𝑥 − 𝑦 = 15 5x−y=15 Answer 1 Subtract the first equation from the second: ( 5 𝑥 − 𝑦 ) − ( 3 𝑥 − 𝑦 ) = 15 − 7 (5x−y)−(3x−y)=15−7 2 𝑥 = 8 2x=8 𝑥 = 4 x=4 Substitute 𝑥 = 4 x=4 into the first equation: 3 ( 4 ) − 𝑦 = 7 3(4)−y=7 12 − 𝑦 = 7 12−y=7 𝑦 = 5 y=5 So, the solution is ( 4 , 5 ) (4,5). Question 4 The Richter scale reading for an earthquake measures its intensity 𝐼 I as: 𝐼 = 1 0 𝑅 I=10 R A quake measuring 𝑅 = 8.5 R=8.5 on the Richter scale has an intensity: Exponential form: 1 0 8.5 10 8.5 Radical form: 1 0 17 10 17 Thus, the intensity is: 1 0 8.5 and 1 0 17 10 8.5 and 10 17 Question 5 Rewrite the equation for this circle in standard form: 𝑥 2 − 16 𝑥 + 𝑦 2 − 10 𝑦 + 73 = 0 x 2 −16x+y 2 −10y+73=0 Answer 1 Complete the square for 𝑥 x and 𝑦 y: Rewrite as: ( 𝑥 2 − 16 𝑥 ) + ( 𝑦 2 − 10 𝑦 ) = − 73 (x 2 −16x)+(y 2 −10y)=−73 For 𝑥 x-terms: 𝑥 2 − 16 𝑥 → ( 𝑥 − 8 ) 2 − 64 x 2 −16x→(x−8) 2 −64 For 𝑦 y-terms: 𝑦 2 − 10 𝑦 → ( 𝑦 − 5 ) 2 − 25 y 2 −10y→(y−5) 2 −25 Substitute back: ( 𝑥 − 8 ) 2 − 64 + ( 𝑦 − 5 ) 2 − 25 = − 73 (x−8) 2 −64+(y−5) 2 −25=−73 ( 𝑥 − 8 ) 2 + ( 𝑦 − 5 ) 2 = 42 (x−8) 2 +(y−5) 2 =42 The standard form is: ( 𝑥 − 8 ) 2 + ( 𝑦 − 5 ) 2 = 42 (x−8) 2 +(y−5) 2 =42 The center is ( 8 , 5 ) (8,5) and the radius is 42 42 .