Why Study Math? Linear Systems and the Substitution Method - Part I

After having read my previous articles on lines and their representative equations, we are now going to look at these objects two at a time, in which case we have what is known as a linear system. When we have two lines, and we try to graph them on the same coordinate plane, three things can happen: one, the lines may never intersect, in which case they are parallel; two, the lines may coincide, in which case they are identical; or three, the lines can intersect in one point, the unique solution of the system. Here we are going to look at how to find that unique solution using the method known as substitution.

In my article, "Why Study Math? - Linear Equations and Slope-Intercept Form," we talked about the standard form for the equation of the line, that is Ax + By = C. When we have a linear system, or pair of lines, whose solution we are looking to find, we usually write them in standard form. One of the advantages of standard form is that it allows us to solve the system more easily when we try to eliminate one of the variables. This method, known as elimination or linear combinations, is another method that we will look at in an upcoming article. For now, we are going to focus on the method of substitution, which
works particularly nicely when one of the equations is in slope-intercept form.

The method of substitution, as the name implies, allows us to substitute one of the variables by its value in terms of the other, and then solve for the variable. For example, take the following system: y = 2x + 1 and 3x - 2y = 5. The first equation tells us what y is in terms of the variable x. We use this in the second equation and replace y with 2x + 1. Doing this we obtain,
3x - 2(2x + 1) = 5. Now that we have an equation in x only, we can solve to get x = -7. We then plug this value in the first equation to get y = 2(-7) + 1 or -13. The unique solution of this system then is -7, -13, which is a point and the unique point where, if we were to graph these two lines on the same coordinate plane, the two lines intersect.

Although this method usually gives students trouble, there really is no difficulty with it if you keep your head on straight and your pencil sharpened. For example, take the following system:
3x - 5y = 4, and y = -3x + 10. By using the second equation in the first, we obtain
3x - 5(-3x + 10) = 4. Simplifying, we have 3x + 15x -50 = 4. Rearranging terms, we get
18x = 54 or x = 3. We use this value of x to get y = -3(3) + 10 or y = 1.

There really is no difficulty with this method if you keep your figures neat and precise. Just keep your eraser handy for an error or two and remember that errors are okay because the more mistakes you make, the better you are going to get. Remember: if at first you don't succeed, just do another problem. Happy problem solving.