Suppose I have an equation x+2y=3 and label it (2). And I have a previous equation x+y=2 labeled (1). Now I want to put the two equations together and solve them, what should I write?


I truly believe that the best phrase to say what you described in your question is solving equations, or solving a set of equations, or solving a system of equations.

It is worth mentioning that there are many approaches and methods to solve a system of equations. We can solve it algebraically. We can solve it numerically. We can solve it by using elementary algebra. We can also treat the whole set of equations as a matrix and solve it with a method called row reduction (also known as Gaussian elimination).

However, I usually prefer the simplest choice when I have many. And I would like to quote from this page in Wikipedia,

The simplest kind of linear system involves two equations and two variables:

  enter image description here

One method for solving such a system is as follows. First, solve the top equation for x in terms of y:

  enter image description here

Now substitute this expression for x into the bottom equation:

  enter image description here

This results in a single equation involving only the variable y. Solving gives y = 1, and substituting this back into the equation for x yields x = 3/2. This method generalizes to systems with additional variables (see "elimination of variables" below, or the article on elementary algebra.)

The example above should illustrate how to describe mathematical operations we needed in order to solve a set of equations well enough.

See also:

  • A system of equations is what we called it where I went to school.
    – user230
    Dec 29 '13 at 11:45
  • @snailboat I'm not sure if it's the same in the US. When I was in grade 1-6, our teachers used the words equivalent to solving equation(s). It was hard enough just to solve one equation back then. When I was in grade 7-9, the teachers began to change to use solving a set of equations, after the concept of sets had been introduced. Later, from grade 10 up, the teachers would use the more formal term: solving a system of equations. Dec 29 '13 at 11:54
  • the reference does help a lot. I could always borrow words from those wiki pages. But what word should I use if I'm not so interested in solutions? For example, what should I say if I multiply (1) by 3, subtract (2), and get a third equation 2x+y=3 (without really using the words "multiply" and "subtract"?)
    – arax
    Dec 29 '13 at 17:31
  • When I was at school the phrase for this particular task was to solve simultaneous equations
    – toandfro
    Dec 30 '13 at 0:10

I wanted to give you a few more options that are less formal than the other answers. In a mathematical proof where I can assume that the reader already knows about systems of linear equations and Gaussian elimination, I'll sometimes use the following constructions:

(1) x+y=2
(2) x+2y=3

Using equations (1) and (2), solve for x and y:
Combine equations (1) and (2):

x = 1
y = 1

  • I think combine is the word that I'm looking for, thanks
    – arax
    Dec 29 '13 at 17:36

I don't know in what way you mean by "put the two together". All I can think of is that:

You want to solve the y and x with addition method. Write them like:

x+y=2 (1)

x+2y=3 (2) -

If we add the two equations (or in this case, subtract Eq. 1 from Eq.2), we will get -y = -1, therefore y=1. Next, we substitute 1 in for y into any one of the equations. If we choose to substitute in the first equation, we will get x = 1.

P.S. This is also called elimination method for linear equation. There are some other methods too, like Gaussian Elimination, Substitution, Graphing, Matrices, etc. These methos are usually used for linear equations (linear inequalities). As for the non-linear ones, you have to solve them in other ways. For example, use factoring for quadratic equations,

P.P.S*: Based on Frank's comment, this would form a system of linear equations which can be solved some methods, one of which is addition/elimination.

  • Maybe he means, for example, (1),(2) form a system of linear equations.
    – Yai0Phah
    Dec 29 '13 at 7:50
  • Huh? Really? Is that it then? Thanks (y)
    – Safira
    Dec 29 '13 at 7:52
  • 1
    But it seems that he's looking for some idiomatic and concise expressions. There's a concise verb in my native language. I don't know whether there's something similar in English.
    – Yai0Phah
    Dec 29 '13 at 7:56
  • For non-linear system, is this process still called elimination? For example, I'm using the border condition and the general solution for a differential equation to derive the special solution
    – arax
    Dec 29 '13 at 9:29
  • @FrankScience yeah, that's exactly what I mean. In Chinese it's just "to put together". I think I've read about a English word that have the similar meaning in my textbook, but that's a long time ago and I can't really remember what it is
    – arax
    Dec 29 '13 at 9:34

I would say that the user should equate the two equations.

If (1) x+y=2, then x= 2-y. If (2) x+2y=3, then x=3=2y.

Equating the x's, 2-y=3-2y

Rearranging terms, (2y-y)= 3-2. y=1

Plugging into 1), x=2-y=2-1=1.

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