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**paulb203****Member**- Registered: 2023-02-24
- Posts: 246

I’m told that a linear equation comes in the form y=mx+c where m is the gradient and c is the y intercept. So far so good.

hen I’m told that an example of a practical application of linear equations is as follows.

You’re buying pizza slices and doughnuts for a party. Pizza slices cost 20 rupees. Doughnuts cost 25 rupees. Your budget is 1000 rupees.

You can then use the linear equation 20x+25y=1000 with x being the number of pizza slices and y being the number of doughnuts. So you can have a play around with various ideas for the number of pizza slices you might want (and find out the number of doughnuts you would be able to buy for each suggested number of pizza slices, and vice versa).

But I thought the ‘linear’ part meant ‘produces a straight line when graphed (?).

How does 20x+25y=1000 relate to y=mx+c?

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,365

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**paulb203****Member**- Registered: 2023-02-24
- Posts: 246

Thanks, Jai Ganesh

So it's too simplistic to say that a linear equation comes in the form y=mx+c?

They also come in the forms, point-gradient, general form, etc, etc?

Can they all be rearranged into the gradient-intercept form?

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**KerimF****Member**- From: Aleppo-Syria
- Registered: 2018-08-10
- Posts: 217

I guess we all know that 20x+25y=1000 is equivalent to:

y = -20x/25 + 1000/25

y = -4x/5 + 40 [m=-4/5 and c=40]

But this equation is of a special case. Its 'y' and 'x' have to be positive integers or zero only. That is not all points of the straight line which this linear equation represents are valid as a solution.

Any idea?

Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).

But only a human may have the freedom and ability to oppose his natural robotic nature.

But, by opposing it, such a human becomes no more of this world.

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**Jai Ganesh****Administrator**- Registered: 2005-06-28
- Posts: 47,365

Hi paulb203,

See the links Slope and Finding intercepts from an Equation.

It appears to me that if one wants to make progress in mathematics, one should study the masters and not the pupils. - Niels Henrik Abel.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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**paulb203****Member**- Registered: 2023-02-24
- Posts: 246

KerimF wrote:

I guess we all know that 20x+25y=1000 is equivalent to:

y = -20x/25 + 1000/25

y = -4x/5 + 40 [m=-4/5 and c=40]But this equation is of a special case. Its 'y' and 'x' have to be positive integers or zero only. That is not all points of the straight line which this linear equation represents are valid as a solution.

Any idea?

Thanks, KerimF

I see now that y=-(4/5)x+40, thanks

Can all linear equations be rearranged to the gradient-intercept form?

As for the special case and the straight line etc, I'll need to graph it and get back to you

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**paulb203****Member**- Registered: 2023-02-24
- Posts: 246

Thanks, Jai Ganesh

I'll check those out.

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,512

Can all linear equations be rearranged to the gradient-intercept form?

Yes. Just make y the subject of the equation. If you're not sure how to do that here's a step by step.

1) Multiply out all brackets and multiply by the lowest common denominator to eliminate any fractions.

2) Move terms about so that all the y containing terms are on the left and everything else is on the right.

3) If there's more than one y term, factorise it out so you have y(......) on the left.

4) Divide by the bracket from the line above so that y is on its own.

eg.

2x/3 + 4(x+y) = x - y + 5

1) clear that bracket and times all by 3

2x + 12x + 12y = 3x - 3y + 15

2) move terms

12y + 3y = 3x - 12x - 2x + 15

3) factorise the y. In this case we can do a lot of simplifying as well.

y(12+ 3) = 15y = -11x + 15

4) y = -11x/15 + 1

Bob

check put x = 15 in the final line so y = -11 + 1 = -10

Substitute these values into what we had at the beginning. If it works then I've probably not made an error.

LHS = 10 + 4 times (15-10) = 10 + 4 times 5 = 30

RHS = 15 --10 +5 = 15 + 10 + 5 = 30

This method of checking can be very useful. The rules of algebra are the same as the rules for numbers so if you choose some numbers to fit an equation at one stage in the working, those same numbers should 'fit' at every line in the working. If they don't you've found an error with your working.

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**paulb203****Member**- Registered: 2023-02-24
- Posts: 246

Thanks, Bob, really helpful

Two of the examples of linear equations on the MIF page are;

5x=6

and,

y/2=3

I rearranged them and got,

x=0.12

and,

y=6

Those would produce points on the graph, as opposed to lines, yeah? If so, why are they classed as linear equations, if they're just points?

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**Bob****Administrator**- Registered: 2010-06-20
- Posts: 10,512

5x= 6 means x = 1.2

An equation that gives rise to a line is why the word linear is used.

It's the absence of any powers eg x^2 that allows this to happen.

This usage has spilt over into equations generally so in your examples linear is used because there's no powers .

Bob

Children are not defined by school ...........The Fonz

You cannot teach a man anything; you can only help him find it within himself..........Galileo Galilei

Sometimes I deliberately make mistakes, just to test you! …………….Bob

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**KerimF****Member**- From: Aleppo-Syria
- Registered: 2018-08-10
- Posts: 217

paulb203 wrote:

x=0.12

and,

y=6Those would produce points on the graph, as opposed to lines, yeah? If so, why are they classed as linear equations, if they're just points?

For instance,

x=0.12 produces a vertical line if drawn on the conventional y_x plane. [but this is a special case, |m|=infinity and |c|=infinity]

y=6 also produces a horizontal line on y_x plane. [here, m=0 and c=6]

Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).

But only a human may have the freedom and ability to oppose his natural robotic nature.

But, by opposing it, such a human becomes no more of this world.

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**paulb203****Member**- Registered: 2023-02-24
- Posts: 246

Bob wrote:

5x= 6 means x = 1.2

An equation that gives rise to a line is why the word linear is used.

It's the absence of any powers eg x^2 that allows this to happen.

This usage has spilt over into equations generally so in your examples linear is used because there's no powers .

Bob

Ah, of course; I knew about lines such as x=1.2, and y=6; it was the 5x, and the y/2, and the rearranging that threw me

Thanks.

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**paulb203****Member**- Registered: 2023-02-24
- Posts: 246

KerimF wrote:

paulb203 wrote:x=0.12

and,

y=6Those would produce points on the graph, as opposed to lines, yeah? If so, why are they classed as linear equations, if they're just points?

For instance,

x=0.12 produces a vertical line if drawn on the conventional y_x plane. [but this is a special case, |m|=infinity and |c|=infinity]

y=6 also produces a horizontal line on y_x plane. [here, m=0 and c=6]

Thanks, KerimF

With x=0.12 why does m, the gradient, =infinity? I thought it would equal zero as there is zero slope?

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**KerimF****Member**- From: Aleppo-Syria
- Registered: 2018-08-10
- Posts: 217

paulb203 wrote:

With x=0.12 why does m, the gradient, =infinity? I thought it would equal zero as there is zero slope?

Good question. Here are two ways, I think of, to answer it:

[1]

x=0.12 could also be written as 0*y = 1*x - 0.12

And let us assume that the value '0' here is actually very close to 0 [not exactly 0], like 1/1,000,000,000,000 for example.

We get:

y*1/1,000,000,000,000 = 1*x - 0.12

or

y = 1*x*1,000,000,000,000 - 0.12*1,000,000,000,000

As we see, we have now m=1,000,000,000,000 and c=-120,000,000,000

In other words, we can simplify this process by saying that m=infinity and c=-infinity [while the ratio of their infinities is m/c=-1/0.12]

[2]

As you know, 'm' is the slope of the line, which is defined by the linear equation. Therefore, m = dy/dx.

Since the line is vertical on the x-axis (x=0.12 for any value of y), dx is zero for any value of dy.

m = dy/dx = dy/0 = infinity [as you know, a number divided by zero is equivalent to a very big number, infinity]

Also, 'c' is where the line intercepts the y-axis. Since both, the y-axis and the equation's line, are on the same plane and vertical on the same straight (x-axis), they are parallel. Therefore, they intersect at infinity. Again,'c' here is equal to infinity

Kerim

*Last edited by KerimF (2024-07-07 05:49:20)*

Every living thing has no choice but to execute its pre-programmed instructions embedded in it (known as instincts).

But only a human may have the freedom and ability to oppose his natural robotic nature.

But, by opposing it, such a human becomes no more of this world.

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