# Simultaneous equations

#### Marcus Clayson

##### New member
I came across the following question and as there are no values after an equals sign I am at a loss as to how to even begin to go about solving it.
The question is:
the lengths of the sides of an equilateral triangle are:
3(x+y) cm
(5x+2y-1) cm
5(x+1)cm
I expanded the brackets and combined like terms as, if the triangle is equilateral then all sides must be equal, but as there is no numerical value - e.g. = 12 - I cannot see how to find a numerical value for either x or y.

#### Zermelo

##### Junior Member
Well, the 3 sides are equal, so, you can write out 3 equations from this formulation(a=b, b=c, a=c). You can even get the value of y straight out of one of these equations! Try it out!
On the other hand, have you ever worked with systems of linear equations?

#### Harry_the_cat

##### Elite Member
Well, the 3 sides are equal, so, you can write out 3 equations from this formulation(a=b, b=c, a=c). You can even get the value of y straight out of one of these equations! Try it out!
On the other hand, have you ever worked with systems of linear equations?
I would say Marcus knows about simultaneous equations because of the title of the thread.
You only need two equations, so let the first expression equal the second, and the second equal the third.

#### JeffM

##### Elite Member
As zermelo said, the equal signs are implied by the fact that the triangle is EQUIlateral.

What might be confusing a bit is that Zermelo said you can derive three equations and that Harry said you need two equations. There is no inconsistency. Pick any two of the three.

#### HallsofIvy

##### Elite Member
You have, setting the sides equal in pairs, 3(x+ y)= 5x+ 2y- 1, 5x+ 2y-1= 5(x+1), and 3(x+ y)= 5(x+ 1). Of course, any one of those equations is implied by the other two so you really have two independent equations in x and y.
Say we take the first two.
3(x+ y)= 3x+ 3y= 5x+ 2y- 1 reduces to 2x- y= 1 (subtract 3x and 3y from both sides and add 1 to both sides).
5x+ 2y- 1=5x+ 5 reduces to 2y= 6 (subtract 5x from both sides and add 1 to both sides.

2x- y= 1 and 2y= 6 are particularly easy to solve! And, of course, once you have x and y, plug them into 3(x+ y), 5(x+1), and 5x+ 2y- 1 to verify that the three sides are the same length.

#### Jomo

##### Elite Member
Suppose one side is x + 3 and another side was 2x. Clearly x=3 and both sides are 6. So yes, without numbers on the other side of the equal sign this can be done!