7 pencils and 2 rulers weigh 93 grams. 5 pencils and 6 rulers weigh 135 grams. How much does: a) a pencil weigh
b) a ruler weigh

7p + 2r = 93
5p + 6r = 135
Times the first one by 3 to get an equal number of rulers so you get
21p + 6r = 279
5p + 6r = 135
Take the second one from the first and you should get
16p = 144
Divide 144 by 16 and the the weight of a pencil is 9 grams then you can substitute it into one of the first equations for example
7 times 9 + 2r = 93
63 + 2r = 93
Move the 63 across by taking it away from both sides to get 2r = 30 then divide 30 by two to get 15 which is how much a ruler ways

A) pencil = 9 grams
B) ruler = 15 grams

Answer: f(x) = -2x^3 +3x^2 +11x -6see attachedan infinite number. since the magnitude of the leading coefficient is not specified, it may be any negative number. (we have chosen the smallest magnitude integer that makes all coefficients be integers.)step-by-step explanation:

1. when "a" is a root of a polynomial, (x -a) is a factor of it. for the three roots given, the factors of the desired polynomial are (x +2)(x -1/2)(x -3).

in order to make the leading coefficient be negative, we need to multiply this product by a negative number. any negative number will do, but we choose a small (magnitude) value that will eliminate the fraction: -2.

then

f(x) = -2(x +2)(x -1/2)(x -3) = -(x +2)(2x -1)(x -3)

= -(2x² +3x -2)(x -3)

= -(2x³ -3x² -11x +6)

f(x) = -2x³ +3x² +11x -6

2. a graph created by the desmos on-line graphing calculator is shown, and the zeros are highlighted.

3. as indicated in part 1, the multiplier of this equation can be anything and the zeros will remain the same. you want a negative leading coefficient, so the "anything" is restricted to any of the infinite number of numbers that will make that be the case.

Will show energy on x axis time on the y axis feel free to ask any questions