# Algebraic expression

#### eddy2017

##### Full Member
Hi, sorry for the mix-up.
I have this problem.
Which algebraic expression can be used to determine the sales price. Use the following table.
I am attaching the table here.
I'm studying finding relations from a table and trying to get the hang of it.
This is what i have been able to do from i have studied so far.
I will assign x and y values to the table.
Regular price= x
Sale price= y
Now, I see what is going on with both x and y
X increases by 1
While y also increases by 3.
In other words I am adding 3 to the first number on the y section.
So a possible equation that I could write taking this analysis would be:
I am stuck in how to start writing the equation.
Thanks in advance for any hints.
eddy

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#### eddy2017

##### Full Member
I ended up like you advised. Let's see if it is okay.
y=3x+3
That is a possible algebraic expression that i can write according the table given.

#### Subhotosh Khan

##### Super Moderator
Staff member
I ended up like you advised. Let's see if it is okay.
y=3x+3
That is a possible algebraic expression that i can write according the table given.
You have x = 16 with y = 12 (in your data set).

Does your equation match any of the data-points [(8,6) , (12,9) , (16,12) & (20,15)] ?

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#### Jomo

##### Elite Member
Eddy you have to try!

Your equation, y = 3x+3, clearly states that if you triple x and then add 3 you will get the y value.

When you triple 8 and then add 3 do you get 6? If yes, then maybe you have the correct formula. Check if the other points are satisfied by your equation. If when you triple 8 and add 3 you do not get 6, then your formula is wrong.

So is your formula correct or not?

#### eddy2017

##### Full Member
You have x = 16 with y = 12 (in your data set).

Does your equation match any of the data-points [(8,6) , (12,9) , (16,12) & (20,15)] ?
I got it. Let me try them.

#### eddy2017

##### Full Member
(8, 6) doesn't satisfy it
(12,9)... doesn't satisfy either
like none of the rest.
The expression that I wrote is incorrect.

#### eddy2017

##### Full Member
Give some time to think a little bit more and watch some videos.

#### eddy2017

##### Full Member
I watched something that brought some order to my thinking. Maybe it would help.
What is the change in y as compared to the corresponding change in x

Take a look at the change in y > from 6 to 9 =a change in 3

Now, look at the change in x> from 8 to 12 = a change of 4 (increase)

The next change in y goes from 9 to 12 = an increase of 3 with a corresponding increase of the x value of 4

Then from 9 to 12= we have an in crease of y of 3 while the x has an increase of 4

Then from 12 to 15 there is an increase of 3 in the y and an increase of 4 in the x value from 16 to 20
The change in y as compared to the change in x has a ration of 3:4 ( 3 to 4, 3/4)
up to here, I understand.
A question, because I am not sure if it is needed here.
Would my next step be to write the equation for the slope of a line, the equation of a linear function?.

#### JeffM

##### Elite Member
This will be an indirect answer.

Such questions should come with a BIG WARNING. It is impossible to be sure that you can derive the true function from a partial table. But we almost never have complete information so we frequently want to estimate the function from the information we do have while always understanding we may be wrong. The text books that say something is the right answer are actively misleading students (and possibly some teachers). What the text books usually give are the simplest reasonable estimates.

Now let’s think of a true linear function

$$\displaystyle y = b + mx.$$

$$\displaystyle \therefore x_1 = a \implies y_1 = b + am$$

$$\displaystyle x_2 = 2a \implies y_2 = b + 2am$$

$$\displaystyle \therefore y_2 - y_1 = b + 2am - (b + am) = am.$$

$$\displaystyle x_2 - x_1 = 2a - a = a.$$

$$\displaystyle \dfrac{y_2 - y_1}{x_2 - x_1} = \dfrac{am}{a} = m$$.

Now we do not know in this case whether the function is linear.

But when $$\displaystyle \dfrac{y_k - y_{k-1}}{x_k - x_{k-1}} \approx p$$

for a lot of values of k, then it is a reasonable supposition that f(x) either is q + px or is approximately q + px. Of course we still have to find q.

This is the method of “first differences,” which is a test to see whether the available evidence strongly suggests that the function is linear.

Make sense?

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#### eddy2017

##### Full Member
Eurekaa, I FOUND IT!
Applying the formula for the slope of a line. Holey LEE!

What is the change in y as compared to the corresponding change in x.
Take a look at the change in y > from 6 to 9 =a change in 3
Now, look at the change in x> from 8 to 12 = a change of 4 (increase)
The next change in y goes from 9 to 12 = an increase of 3 with a corresponding increase of the x value of 4

Then from 9 to 12= we have an in crease of y of 3 while the x has an increase of 4
Then from 12 to 15 there is an increase of 3 in the y and and increase of 4 in the x value from 16 to 20

If you take a look at the first change in y as compared to the first change in x we could express it as a ratio of ¾

We can see that the y’s are going up by 3

And the x’s are going up by 4

We can see that the line is going at this constant rate with a slope of ¾ or 0.75
So if we wanna write the equation we would say:
We know that the slope is ¾ so

y= mx + b

y = (3/4)x + b

we don’t know the y-intercept value(the b value), but we can pick one of the coordinates( or ordered pairs), let’s say, (12,9),where y= 9, and x =12

and we can solve for b

y =3/4 + b

9=3/4(12)+ b

9=9+b

b=0

so,

let’s plug in 0 for b

y= mx + b

y= 0.75x + 0

I will check with another ordered pair from the table, let’s say (20,15)

y=0.75(20)+0

y=15

let’s take another (8,6)

y= mx + b

y=0.75(8)+0

y=6

• Subhotosh Khan and JeffM

#### eddy2017

##### Full Member
Eurekaa, I FOUND IT!
Applying the formula for the slope of a line. Holey LEE!

What is the change in y as compared to the corresponding change in x.
Take a look at the change in y > from 6 to 9 =a change in 3
Now, look at the change in x> from 8 to 12 = a change of 4 (increase)
The next change in y goes from 9 to 12 = an increase of 3 with a corresponding increase of the x value of 4

Then from 9 to 12= we have an in crease of y of 3 while the x has an increase of 4
Then from 12 to 15 there is an increase of 3 in the y and and increase of 4 in the x value from 16 to 20

If you take a look at the first change in y as compared to the first change in x we could express it as a ratio of ¾

We can see that the y’s are going up by 3

And the x’s are going up by 4

We can see that the line is going at this constant rate with a slope of ¾ or 0.75
So if we wanna write the equation we would say:
We know that the slope is ¾ so

y= mx + b

y = (3/4)x + b

we don’t know the y-intercept value(the b value), but we can pick one of the coordinates( or ordered pairs), let’s say, (12,9),where y= 9, and x =12

and we can solve for b

y =3/4 + b

9=3/4(12)+ b

9=9+b

b=0

so,

let’s plug in 0 for b

y= mx + b

y= 0.75x + 0

I will check with another ordered pair from the table, let’s say (20,15)

y=0.75(20)+0

y=15

let’s take another (8,6)

y= mx + b

y=0.75(8)+0

y=6
Thanksssssss   #### Jomo

##### Elite Member
Eurekaa, I FOUND IT!
Applying the formula for the slope of a line. Holey LEE!

What is the change in y as compared to the corresponding change in x.
Take a look at the change in y > from 6 to 9 =a change in 3
Now, look at the change in x> from 8 to 12 = a change of 4 (increase)
The next change in y goes from 9 to 12 = an increase of 3 with a corresponding increase of the x value of 4

Then from 9 to 12= we have an in crease of y of 3 while the x has an increase of 4
Then from 12 to 15 there is an increase of 3 in the y and and increase of 4 in the x value from 16 to 20

If you take a look at the first change in y as compared to the first change in x we could express it as a ratio of ¾

We can see that the y’s are going up by 3

And the x’s are going up by 4

We can see that the line is going at this constant rate with a slope of ¾ or 0.75
So if we wanna write the equation we would say:
We know that the slope is ¾ so

y= mx + b

y = (3/4)x + b

we don’t know the y-intercept value(the b value), but we can pick one of the coordinates( or ordered pairs), let’s say, (12,9),where y= 9, and x =12

and we can solve for b

y =3/4 + b

9=3/4(12)+ b

9=9+b

b=0

so,

let’s plug in 0 for b

y= mx + b

y= 0.75x + 0
Write the function as y = .75x (or y= 3x/4). That is no need to write the +0

Good job!

#### eddy2017

##### Full Member
Great!. Thanks to you all.

• Jomo