Consider the equation 2x + 2y = 8.
part a: write an equivalent equation by multiplying the equation by 3.
does the equation have the same solution set?
part b: write an equivalent equation by dividing the equation by 2. does
the equation have the same solution set?

Part A) see the explanation

Part B) see the explanation

Step-by-step explanation:

we have

Part A) Write an equivalent equation by multiplying the equation by 3.

Does the equation have the same solution set?

Multiply by 3 both sides

Remember that

The Multiplication Property of Equality states that if you multiply both sides of an equation by the same number, the sides remain equal

therefore

The equivalent equation have the same solution set by multiplication property of equality

Part B) Write an equivalent equation by dividing the equation by 2. Does

the equation have the same solution set?

Divide by 2 both sides

Remember that

The Division Property of Equality states that if you divide both sides of an equation by the same number, the sides remain equal

therefore

The equivalent equation have the same solution set by division property of equality

step-by-step explanation:

part a. the differences are decreasing, so we know the sequence is not arithmetic. the ratios are constant (=3/4), so we know the sequence is geometric.

part b. from part a, we know the common ratio is 3/4, so each term is 3/4 of the one before. the initial term is the number of gallons at h=1.

g[h] = (3/4)·g[h-1]

g[1] = 384 . . . . initial value required for a recursive definition

part c. the general formula for a geometric sequence is

a[n] = a[1]·r^(n-1)

we have a[1] = 384, r = 3/4, and the term number given by h. the formula that follows is

g[h] = 384·(3/4)^(h-1)

part d. filling in the value and doing the arithmetic, we have

g[12] = 384·(3/4)^11 ≈ 384·0.042235136 ≈ 16.2183 . . gallons

part e. each gallon is 4 quarts, so

16.2183 gal × 4 qt/gal = 64.8732 qt

answer: cos [   arcsin (3/5) - arccos (1/2) ] =

step-by-step explanation: × = 3/5 = 0.6--> x = 36.87 and x = 180 - 36.87 = 143.13 deg

cos y = 1/2 --> y = +- 60 deg a. cos (x - y) = cos (143.13 - 60) = cos 83,13  

=

0.12

b. cos (36.87 - 60) = cos (-23.23) = 0.92