# How can one straight line have two equations?

For a question in my maths homework I got the answer y=-0.666x +4, but the answers tell me it's 2x+3y=12.

how can this be?

### 16 Answers

- Some BodyLv 71 year agoFavorite Answer
So you got:

y = -2/3 x + 4

Multiply both sides by -3:

-3y = 2x - 12

Rearrange:

2x + 3y = 12

Both equations are the same, just written differently.

- Jeff AaronLv 71 year ago
Those two lines are close but not exactly the same.

If you multiply the first equation by 3, you get:

3y = -1.998x + 12

Add 1.998x to both sides:

1.998x + 3y = 12

If -0.666 was rounded off from -2/3, then in fact the two lines are the same.

- AmyLv 71 year ago
Taking a simpler line than your problem, consider the equation y = x.

This line goes through the points (1,1), (3,3), (-2.7, -2.7), etc.

The "equation of a line" is a description of what paired values of x and y describe the set of points that are on the line. An equation is a statement that can be either true or false.

y = x is TRUE when x = 3 and y = 3.

It is false when x = 3 and y = 4.

Therefore the point (3,3) is on the line y=x, and the point (3,4) is NOT on the line y=x.

Now consider the line 2y = 2x.

You'll find that it describes the exact same set of points. This equation statement is true at all the same (x,y) values where y=x is true, and false at all the same values where y=x is false.

In general, multiplying both sides of an equation by the same number (except 0) will produce a different equation that has the same truth values.

y=x is the same as 2y=2x is the same as 17y=17x.

Adding the same number to both sides of an equation, or pretty much any mathematical manipulation done to both sides, produces an equation that describes the same line.

y = x-1 is the same as y+1 = x is the same as y - 3 = x - 4.

This is an important principle for solving algebraic equations.

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- Anonymous1 year ago
2x+3y=12.

3y=-2x+12

y=(-2/3)x+4

I assume you meant to post this and not y=-0.666x +4

-2/3 is not the same as -.666

- Jeffrey KLv 61 year ago
They are the same equation.

y = -.666x + 4

multiply by 3

3y = -2x + 12

2x + 3y = 12

- RealProLv 71 year ago
4 = 2 + 2 but also 4 = 1 + 3 so are you surprised about that as well? Cause that's basically the equivalent of what you're asking.

At this point, you should know how to REARRANGE EQUATIONS already.

Realize that a line is a set of points that have two coordinates (numbers) in order: x and y.

A point is a part of the line if its x and y satisfy the equation of the line when you put them inside it.

But then when you do the SAME THING to both sides of the equation, that does not change the equation. So the SAME pair of numbers x and y will STILL satisfy it.

y = (-2/3)x + 4

Multiply by 3 on BOTH sides

3y = -2x + 12

Add 2x to BOTH sides

2x + 3y = 12

Two steps.

Also

x = 6 - 1.5y

2x + 3y - 12 = 0

x/6 + y/4 = 1

2000x + 3000y = 12,000

So that's actually more than just 2.

Please try to understand you can't claim equality out of laziness.

2/3 = 0.666666666666666... forever. You can't stop at THREE sixes and say it's the same. At least say 0.666...

- az_lenderLv 71 year ago
Sure, that's the same thing.

Multiply "your" equation by 3, you get

3y = -2x + 12, equivalent to

2x + 3y = 12.

The only thing "wrong" with yours is that you rounded off an inexact value, when the 0.666 probably should have been

0.66666666666666666 et cetera

- SamwiseLv 71 year ago
There are an infinite set of equivalent equations for any straight line.

The key here is that the equations have to be equivalent,

something we can check with a little algebra.

2x + 3y = 12

3y = -2x + 12 [subtracting 2x from each side]

y = -(2/3)x + 4 [dividing each side by 3]

Taking your decimal fraction -0.666 as an approximation of -2/3,

the two equations are equivalent

[though, correctly rounded, it should have been -0.667,

and generally I'd prefer the exact fraction to the decimal approximation].

Were you told to give the answer in "standard form"?

Your answer [again, allowing for that decimal approximation]

is in slope-intercept form, not standard form.

- RaymondLv 71 year ago
The same equation can be written in different ways.

Sometimes, for homework, you are asked (or expected) to write your answer in a specific way.

One format is called the "slope intercept"

y = mx + c

where m is the slope (how fast y increases when x goes up by 1)

and c is the intercept (the value of y, when x=0)

Yours appear to be

y = (-2/3)x + 4

From there, you can go to any of the other "standard" formats.

Multiply both sides by 3

3y = -2x + 12

Add 2x to both sides

2x + 3y = 12

(this is, apparently, the format you were expected to use)

Subtract 12 from both sides

2x + 3y - 12 = 0

(This is also known as a standard format).

If anything, you try to avoid fractions (if you can); also, if you are stuck with using a fraction, you try to make it a ratio of integers

(I suspect that your 0.666 was meant to represent 2/3, which is really 0.66666666666... forever)

In other words, 0.666 is not quite the same as 2/3.

- derframLv 71 year ago
Convert your answer into 'Standard' form

y = -0.666x + 4; multiply both sides of your equation by 3

3y = -2x + 12; add 2x to both sides

2x + 3y = 12; equation in 'standard' form

[(2/3) = 0.666...ad inf.] ad inf stands for ad infinitum, latin for forever. You don't need a string of

6's which labels you as an amateur.