Linear equations and graphs #15 - Questions and Answers

Question 1 The constant differences of an 8-degree polynomial is 241,920. What is the leading coefficient? Answer The relationship between the constant difference 𝑑 d, the leading coefficient 𝑐 c, and the degree, 𝑛 n, of a polynomial is 𝑐 = 𝑑 𝑛 ! c= n! d ​ . If the polynomial has a degree of 𝑛 = 8 n=8, a constant difference of 𝑑 = 241 , 920 d=241,920, and a leading coefficient of 𝑎 a, then: 241 , 920 = 𝑎 ⋅ 8 ! 241,920=a⋅8! The definition of a factorial is: 8 ! = 8 ⋅ 7 ⋅ 6 ⋅ 5 ⋅ 4 ⋅ 3 ⋅ 2 ⋅ 1 = 40 , 320 8!=8⋅7⋅6⋅5⋅4⋅3⋅2⋅1=40,320 Using this definition gives 8 ! = 40 , 320 8!=40,320. We then have: 241 , 920 = 𝑎 ⋅ 40 , 320 241,920=a⋅40,320 Dividing both sides by 40,320 gives a leading coefficient of: 𝑎 = 241 , 920 40 , 320 = 6 a= 40,320 241,920 ​ =6 Question 2 How to complete the table with the equation 𝑦 = 10 𝑥 − 4 y=10x−4? Answer To fill in a table for an equation, you start with the 𝑥 x's. Imagine your table already has them filled in. Let’s say that 𝑥 = 1 x=1 is in your table. Then our job is to find the corresponding 𝑦 y-value. We do this by "substituting" the value 𝑥 = 1 x=1 into the equation. Let’s give that a try. 𝑦 = 10 𝑥 − 4 y=10x−4 𝑦 = 10 ( 1 ) − 4 y=10(1)−4 𝑦 = 10 − 4 y=10−4 𝑦 = 6 y=6 So in the row with 𝑥 = 1 x=1, you’d put 𝑦 = 6 y=6. Let’s try a different simple number. How about 𝑥 = 0 x=0? We’ll substitute it in and simplify in exactly the same way. 𝑦 = 10 𝑥 − 4 y=10x−4 𝑦 = 10 ( 0 ) − 4 y=10(0)−4 𝑦 = 0 − 4 y=0−4 𝑦 = − 4 y=−4 So that 𝑦 y-value ends up being − 4 −4. Question 3 Which equation must be true? A) 𝑥 − 𝑦 = 𝑥 − 𝑦 x−y=x−y B) 𝑚 + 𝑝 = 𝑝 + 𝑚 m+p=p+m C) ( 𝑑 − 𝑒 ) − 𝑓 = ( 𝑑 − 𝑓 ) − 𝑒 (d−e)−f=(d−f)−e D) ( 𝑟 + 𝑠 ) + 𝑡 = 𝑟 + ( 𝑠 + 𝑡 ) (r+s)+t=r+(s+t) Answer Subtraction is not commutative. For example, 3 − 2 = 1 3−2=1, but 2 − 3 = − 1 2−3=−1. The equation 𝑥 − 𝑦 = 𝑦 − 𝑥 x−y=y−x must then be false. Division is also not commutative. For example, 4 ÷ 2 = 2 4÷2=2, but 2 ÷ 4 = 0.5 2÷4=0.5. The second equation 𝑚 + 𝑝 = 𝑝 + 𝑚 m+p=p+m must then be false. Subtraction is also not associative. Distributing the negative on the right side of the equation ( 𝑑 − 𝑒 ) − 𝑓 ≠ ( 𝑑 − 𝑓 ) − 𝑒 (d−e)−f  =(d−f)−e. The third equation is then false. Addition is associative. The Associative Property of Addition states that ( 𝑎 + 𝑏 ) + 𝑐 = 𝑎 + ( 𝑏 + 𝑐 ) (a+b)+c=a+(b+c). Therefore ( 𝑟 + 𝑠 ) + 𝑡 = 𝑟 + ( 𝑠 + 𝑡 ) (r+s)+t=r+(s+t). The correct answer is then choice D. Question 4 Simplify each expression: − 𝑛 + 9 𝑛 + 3 − 8 − 8 𝑛 −n+9n+3−8−8n 3 ( − 4 𝑥 + 5 𝑦 ) − 3 𝑥 ( 2 + 4 ) 3(−4x+5y)−3x(2+4) 5 − 4 𝑦 + 2 + 9 𝑦 5−4y+2+9y − 2 𝑥 + 3 𝑦 − 5 𝑥 − ( − 8 𝑦 ) −2x+3y−5x−(−8y) Answer − 𝑛 + 9 𝑛 + 3 − 8 − 8 𝑛 = ( − 𝑛 + 9 𝑛 − 8 𝑛 ) + ( 3 − 8 ) −n+9n+3−8−8n=(−n+9n−8n)+(3−8) = ( 0 𝑛 ) − 5 = − 5 =(0n)−5=−5 3 ( − 4 𝑥 + 5 𝑦 ) − 3 𝑥 ( 2 + 4 ) 3(−4x+5y)−3x(2+4) = − 12 𝑥 + 15 𝑦 − 3 𝑥 ( 6 ) =−12x+15y−3x(6) = − 12 𝑥 + 15 𝑦 − 18 𝑥 = − 30 𝑥 + 15 𝑦 = 15 ( − 2 𝑥 + 𝑦 ) =−12x+15y−18x=−30x+15y=15(−2x+y) 5 − 4 𝑦 + 2 + 9 𝑦 = ( 5 + 2 ) + ( − 4 𝑦 + 9 𝑦 ) 5−4y+2+9y=(5+2)+(−4y+9y) = 7 + 5 𝑦 =7+5y − 2 𝑥 + 3 𝑦 − 5 𝑥 − ( − 8 𝑦 ) −2x+3y−5x−(−8y) = − 2 𝑥 + 3 𝑦 − 5 𝑥 + 8 𝑦 =−2x+3y−5x+8y = ( − 2 𝑥 − 5 𝑥 ) + ( 3 𝑦 + 8 𝑦 ) =(−2x−5x)+(3y+8y) = − 7 𝑥 + 11 𝑦 =−7x+11y Question 5 63 : 27 = 49 : 𝑥 63:27=49:x Answer Let 𝑥 x be the missing number. We can write the proportion as: 63 27 = 49 𝑥 27 63 ​ = x 49 ​ The left side can be reduced to lowest terms since both numbers have 9 as their factor: 63 27 = 9 ⋅ 7 9 ⋅ 3 = 7 3 27 63 ​ = 9⋅3 9⋅7 ​ = 3 7 ​ So, 7 3 = 49 𝑥 3 7 ​ = x 49 ​ Cross multiply: 7 𝑥 = 3 ( 49 ) 7x=3(49) 7 𝑥 = 147 7x=147 Divide both sides by 7: 𝑥 = 21 x=21