Example:

(1) 2x + y = 3
(2) x - y = 7

In this example there are two unknown variables, x and y. When solving these two linear equations, we are trying to find what value of x and y satisfy both these equations (x and y are what we are trying to work out).

Here we can see equation (1) and equation (2) have the same y coefficients (the constant number the variable x/y is multiplied by) however one is negative and the other is positive. If we then add equation (1) and equation (2) together, we will eliminate the y variable.

(1) + (2) gives us:

2x + y + x - y = 3 + 7

Now we have eliminated y. Simplifying both sides, we end up with:

3x = 10

To determine the value of the x variable, we need to divide both sides by 3:

x = 10/3

To finish solving the original two simultaneous equations, we should substitute our x value into either equation (1) or equation (2). In this instance, I am going to choose to substitute x into the first equation.

2(10/3) + y = 3
20/3 + y = 3

Subtracting 20/3 from both the left and the right hand side of our equation to maintain the balance gives us our y value…

20/3 + y - 20/3 = 3 - 20/3
y = -11/3

Our answer then is x = 10/3 and y = -11/3.

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Another way we can write this, is as the coordinate (10/3, -11/3). Why would we write this as a coordinate? Remember at the beginning of this post, I gave the definition of a linear equation to be a straight line… well, solving linear equations gives us the point at which the two straight lines meet. We can therefore write this as a coordinate.

Visualising this below, you can see the red line (equation 1) and blue line (equation 2) meeting at the point (10/3, -11/3).