7. Is it proportional? 8. If so, what is the constant of
proportionality? (unit rate)
9. Write an equation that
expresses the table in the
form
Y= KX

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:

Yes it is. Y=2.5x

Step-by-step explanation:

So, lets start with bthe first values of the table and work our way down:

2x=5y

Lets just divide both sides by the 2:

x=5/2y

This means that for every 1 x, there should be 5/2y's.

Or, for every 1 x, there should be 2.5ys.

Lets check the next few to see if thats correct:

3x=7.5y

Divide both sides by 3:

x=7.5/3

This means for every one x, there are again, 2.5ys.

Lets check one more just to be sure:

4x=10y

Divide both sides by 4:

x=10/4

This means for everyone one x, y equals 2.5ys, yet again,

So we can conclude that the poprotional relationship between x and y is:

When x equals 1, y=2.5

Or

Y=2.5x

Hope this helps!

a) $ 9.50

b) 17

c) y-intercept (0, 12). this is the initial amount of money that joe had prior to playing any game.

d) x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e) slope is -0.5. joe's balance will reduce by 0.5 for every game that he plays.

f)

domain; [0,24]

range; [0,12]

g) x + 2y = 24

step-by-step explanation:

a)

if joe plays 5 games, this will imply that x = 5. substitute x = 5 in the equation and solve for y;

y = -.50x + 12

y = -0.50(5) + 12

y = 9.50

b)

if joe leaves the arcade with $3.50, this will mean that y = 3.50. plug in y = 3.50 in the given equation and solve for x;

y = -.50x + 12

3.50 = -0.50x + 12

0.50x = 12 - 3.50

0.5x = 8.5

x = 17

c)

the function given is   y = -.50x + 12.

the y-intercept is the point where the graph of this function will intersect the y-axis. at this point, the value of x is usually 0. plug in x = 0 in the given equation and determine y;

y = -0.50(0) + 12

y = 12

the y-intercept is thus (0, 12). this is the initial amount of money that joe had prior to playing any game.

d)

the x-intercept refers to the point where the graph of a function intersects with the x-axis. at this point, the value of y is 0. substitute y = 0 in the given equation and solve for x;

0 =   -.50x + 12

0.5x = 12

x = 24

x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e)

the slope of the given equation is -0.5. the equation has been given in slope-intercept form. thus the slope is simply the coefficient of x.

joe's balance will reduce by 0.5 for every game that he plays. this is the interpretation of the slope.

f)

domain is the set of x values for which the function is real and defined. the domain will thus be; [0,24]. this will restrict the balance to only assume positive values.

on the other hand, the range is the set of y values for which a function is real and defined. the range in our case will be [0,12]. joe can have between $0 and $12 balance.

g)

the standard form of the equation of a straight line is;

ax + by = c

where a, b, and c are integers.

our equation is given as;

y = -.50x + 12

re-writing the equation we have;

0.5x + y = 12

divide all through by 0.5;

x + 2y = 24