I need a linear equation

SCORPION-xisa [38]

Answer:

A unit rate is the rate of change in a relationship where the rate is per 1.

The rate of change is the ratio between the x and y (or input and output) values in a relationship. Another term for the rate of change for proportional relationships is the constant of proportionality.

If the rate of change is yx, then so is the constant of proportionality. To simplify things, we set yx=k, where k represents the constant of proportionality.

If you solve a yx=k equation for y, (like this: y=kx), it is called a direct variation equation. In a direct variation equation, y varies directly with x. When x increases or decreases, y also increases or decreases by the same proportion.

To find y in a direct variation equation, multiply x by the constant of proportionality, k.

For example: Given the relationship y=7x, the constant of proportionality k=7, so if x=3, then y=3×7 or 21.

Given the same relationship, if x=7, then y=7×7, or 49.

Step-by-step explanation:

4 0

2 years ago

Read 2 more answers

The plane x + y + z = 12 intersects paraboloid z = x^2 + y^2 in an ellipse.(a) Find the highest and the lowest points on the ell

emmasim [6.3K]

Answer:

a)

Highest (-3,-3)

Lowest (2,2)

b)

Farthest (-3,-3)

Closest (2,2)

Step-by-step explanation:

To solve this problem we will be using Lagrange multipliers.

a)

Let us find out first the restriction, which is the projection of the intersection on the XY-plane.

From x+y+z=12 we get z=12-x-y and replace this in the equation of the paraboloid:

$\bf 12-x-y=x^2+y^2\Rightarrow x^2+y^2+x+y=12$

completing the squares:

$\bf x^2+y^2+x+y=12\Rightarrow (x+1/2)^2-1/4+(y+1/2)^2-1/4=12\Rightarrow\\\\\Rightarrow (x+1/2)^2+(y+1/2)^2=12+1/2\Rightarrow (x+1/2)^2+(y+1/2)^2=25/2$

and we want the maximum and minimum of the paraboloid when (x,y) varies on the circumference we just found. That is, we want the maximum and minimum of

$\bf f(x,y)=x^2+y^2$

subject to the constraint

$\bf g(x,y)=(x+1/2)^2+(y+1/2)^2-25/2=0$

Now we have

$\bf \nabla f=(\displaystyle\frac{\partial f}{\partial x},\displaystyle\frac{\partial f}{\partial y})=(2x,2y)\\\\\nabla g=(\displaystyle\frac{\partial g}{\partial x},\displaystyle\frac{\partial g}{\partial y})=(2x+1,2y+1)$

Let $\bf \lambda$ be the Lagrange multiplier.

The maximum and minimum must occur at points where

$\bf \nabla f=\lambda\nabla g$

that is,

$\bf (2x,2y)=\lambda(2x+1,2y+1)\Rightarrow 2x=\lambda (2x+1)\;,2y=\lambda (2y+1)$

we can assume (x,y)≠ (-1/2, -1/2) since that point is not in the restriction, so

$\bf \lambda=\displaystyle\frac{2x}{(2x+1)} \;,\lambda=\displaystyle\frac{2y}{(2y+1)}\Rightarrow \displaystyle\frac{2x}{(2x+1)}=\displaystyle\frac{2y}{(2y+1)}\Rightarrow\\\\\Rightarrow 2x(2y+1)=2y(2x+1)\Rightarrow 4xy+2x=4xy+2y\Rightarrow\\\\\Rightarrow x=y$

Replacing in the constraint

$\bf (x+1/2)^2+(x+1/2)^2-25/2=0\Rightarrow (x+1/2)^2=25/4\Rightarrow\\\\\Rightarrow |x+1/2|=5/2$

from this we get

and the candidates for maximum and minimum are (2,2) and (-3,-3).

Replacing these values in f, we see that

f(-3,-3) = 9+9 = 18 is the maximum and

f(2,2) = 4+4 = 8 is the minimum

b)

Since the square of the distance from any given point (x,y) on the paraboloid to (0,0) is f(x,y) itself, the maximum and minimum of the distance are reached at the points we just found.

We have then,

(-3,-3) is the farthest from the origin

(2,2) is the closest to the origin.

3 0

3 years ago

Which function is most likely graphed on the coordinate plane below?

Ber [7]

Answer:

2nd one

Step-by-step explanation:

6 0

2 years ago

A store is having a sale on walnuts and chocolate chips. For 7 pounds of walnuts and 9 pounds of chocolate chips, the total cost

Jlenok [28]

Answer:

Walnuts cost $1.75, chocolate chips cost $2.75

Step-by-step explanation:

<h3>Let the costs be:</h3>

Walnuts - x
Chocolate chips - y

<h3>Set equations as per question</h3>

For 7 pounds of walnuts and 9 pounds of chocolate chips, the total cost is $37:

7x + 9y = 37

For 5 pounds of walnuts and 3 pounds of chocolate chips, the total cost is $17:

5x + 3y = 17

<h3>Solve the system by elimination</h3>

Multiply the second equation by 3 and subtract the first equation, solve for x:

3(5x + 3y) - (7x + 9y) = 3(17) - 37
15x + 9y - 7x - 9y = 51 - 37
8x = 14
x = 14/8
x = 1.75

Find the value of y:

7*1.75 + 9y = 37
12.25 + 9y = 37
9y = 37 - 12.25
9y = 24.75
y = 24.75/9
y = 2.75

6 0

1 year ago

What is the answer for number 15 16 17

stira [4]

15.

6 pachages
each pcage has 30
6*30=180 total

17 left over
how many used
used=total-left over
used=180-17
used=163

16. xy is 2 digit number
multiplules of 10 (whole multipules) always end with 0 so y=0

and has 3 as one of its factors
factors of 3 is 3,6,9,12
30,60,90 is answer

17.
30 in 9 parks
30 times 9
30=10 times 3
3*10*9
group the 3 and 9
10*(3*9)
3*9=27
10*(27)
270

15. 163
16. 30,60,90
17. see above, 270

6 0

3 years ago

I need a linear equation￼

I need a linear equation