Solve for y in terms of x. x=(y+2)/(y+3)

1 answer:

7 0

Multiply both sides by y+3 to get xy+3x=y+2. Subtract 2 from both sides as well as xy, getting 3x-2=y-xy. After that, divide y from both sides to get (3x-2)/y=1-x. After that, divide both sides by (3x-2) to get 1/y=(1-x)/(3x-2). After that, multiply both sides by y to get 1=y(1-x)/(3x-2), and then multiply both sides by
(3x-2)/(1-x) to get (3x-2)/(1-x)=y

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Among all pairs of numbers (x,y)such that 6x+2y=36 find the pair for which the sum of squares, x^2+y^2 is minimum.

lora16 [44]

First, we simplify 6x+2y=36 into 3x+y=18 by dividing by 2. This means that y=-3x+18.

The sum $x^2+y^2$ can be written as: $x^2+y^2=(x+y)^2-2xy$ ,
<span>
from the binomial expansion formula: </span> $(x+y)^2=x^2+2xy+y^2$ .
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Thus, substituting </span>y=-3x+18 and simplifying we have<span>

</span> $x^2+y^2=(x+(-3x+18))^2-2x(-3x+18)$ $=(-2x+18)^2+6x^2-36x=4x^2-72x+18^2+6x^2-36x$ $=10x^2-108x+18^2$ .

This is a parabola which opens upwards (the coefficient of x^2 is positive), so its minimum is at the vertex. To find x, we apply the formula -b/2a. Substituting b=-108, a=10, we find that x is 108/20=5.4.

At x=5.4, the expression $10x^2-108x+18^2$ , which is equivalent to $x^2+y^2$ , takes it smallest value.

Substituting, we would find $10x^2-108x+18^2=10(5.4)^2-108(5.4)+18^2=291.6-583.2+324$ =32.4 This is the smallest value of the expression.

For x=5.4, y=-3x+18=-3(5.4)+18=1.8.

Answer: (5.4, 1.8)

8 0

2 years ago

The population of a city has increased by 36% since it was last measured. If the current population is 71,400,what was the previ

arsen [322]

Answer:

52,500

Step-by-step explanation:

We can convert the word problem into an equation:

1.36x= 71,400

x represents the number of people in the previous population

We divide both sides by 1.36

x=52,500