There are several ways to solve systems of linear equations. The most common methods are by graphing, elimination, and substitution. Let's start off with one of the most basic methods, graphing. --------------- Graphing Method --------------- 2x + y = 3 3x + 2y = 6 In order to solve this system using the graphing method, we first have to change the two equations into slope-intercept form. 2x + y = 3 --> y = -2x + 3 3x + y = 7 --> y = -3x + 7 Then, we graph these two lines. (Attached Below) The solution set of a system of linear equations when graphing is actually the point at which the two lines intersect. So by graphing the two lines, we can obviously see that the solution set of this problem is (4, -5). --------------- Elimination Method --------------- The concept of elimination revolves around the concept of adding two equations. Using an example, let's see what happens when you add equations together. 2x + y = 3 3x + 2y = 6 ----------- 5x + 3y = 9 Do you see how this works? Now, let's say that the orientation of these two equations were different. What would you do then? 2x + y = 3 6 - 3x = 2y If this situation occurs, you have to rearrange it in a way that the form of the equations match. For example, if you have one in standard form, you have to algebraically return the other equation to the same form. 2x + y = 3 6 - 3x = 2y --> 6 = 3x + 2y --> 3x + 2y = 6 Now that the equations are in the same form, we can begin to solve. However, let's start with a simpler system to demonstrate the concept. 2x - y = 5 3x + y = 5 The process of elimination involves adding equations in a way that one of the unknown variables disappears. In this first example, let's see what happens when we simply add them right away. 2x - y = 5 3x + y = 5 --------- 5x + 0 = 10 Using the Identity Property of Addition, we can simply get rid of the 0. 5x = 10 Now it's all algebra. 5x = 10 x = 10/5 x = 2 Now we have one of the variables. You may be thinking "what about the other variable?". Well, in order to solve for the variable, all you have to do is plug what you have for this variable back into the equation like so: 3x + y = 5 3(2) + y = 5 6 + y = 5 y = -1 It doesn't matter which equation we plug it into because you will get the same result.