Math Help

[Leahbee5]

Without graphing, determine whether the following system would have one solution, no solution, or an infinite number of solutions.

3x + y = 6

2x - y = 3/2

[factor-fiction]

Surely the solution would be where the two straight lines cross, methyl

[bibblebub]

For a pair of simultaneous equations in 2 variables there will always be one solution unless one equation is simply a scaled version of the other (i.e. each term multiplied by the same factor), in which case there is an infinity of solutions.

[jim360]

Or, of course, no solution if the equations are inconsistent. For exmaple, the set of equations

x + y = 2
x + y = 3

Is a pair of simulataneous equations with no solution.

[bibblebub]

Sorry there is an exception to that last part - if you rearrange them both into the form y = Ax+B and A is the same in both then you have parallel lines if you plot them and so there is no solution.

[factor-fiction]

I should have added that if the lines had been parallel (which they aren't), then there would be no solution.

y= -3x +6
y= 2x - 1.5
5x= 7.5
x=1.5
y=1.5

[bibblebub]

jim beat me to it

[Prudie]

But they are therefore not simultaneous equations, just 2 statements which refer to a different pair of values of x and y

[jim360]

No, it does still count as a set of simultaneous equations, even if there is no solution. Even that fact can be significant, depending on where the equations came from.

[zebo]

Add the equations together and get

3x + 2x + y - y = 6 + 3/2
5x = 7 1/2 ( 7.5)
x = 1.5

Don't know if you wanted a solution! As has been said, because the equations are not multiples of each other, there has to be exactly 1 solution, as the equations are linear, the highest power of x is 1,

[Prudie]

No I disagree, simultaneous equations by their definition need variables that satisfy all equations. If they don't they are totally unconnected as in your 2 examples, two lines that have no effect on each other, 2 equations that are not simultaneous.

[factor-fiction]

Anyway, it's after 4pm so this discussion has come too late for Leahbee5

[jim360]

It's not a matter of disagreeing or not, it's about definition. The definition of simultaneous equations is that they are "equations that are considered simultaneously" -- or, more precisely, considered as a set, or system, of equations. The properties of their solution -- or indeed, even whether or not they have one -- are to be determined later.

[Prudie]

And that's not my definition " two or more equations, each containing two or more variables whose values can simultaneously satisfy both or all the equations in the set, the number of variables being equal to or less than the number of equations in the set."
So we'll disagree, as factor said, leahbee could't give an xy.

[jim360]

That definition would be unique in the mathematical community. And the fact is that sometimes a set of simultaneous equations having no solution is the most significant case of all -- so it's worth knowing that it exists as a possibility.