How to Solve Linear Systems

By C.D. Crowder

Linear systems require you to solve for x and y coordinates. By using two related equations, you are able to find the coordinates and create graphs to represent your linear systems. Anyone with a good grip on Algebra will find linear systems relatively simple after a few practice problems.

Write down your two equations on a piece of paper. Also, draw a basic graph on your graphing paper. You will need this later. Until you know how large or small the numbers on the graph will be, do not label the units.

Decide whether you wish to solve for x or y first. Typically, you will solve for x first. This how to will use the two equations x – y = 1 and 6x – 5y = 0.

Choose your base equation. You will be substituting one equation into the other. This how to will use x – y = 1 as the base.

Solve your second equation for x. Remember if you do an operation on one side of the equation, you must repeat it on the other side of the equation.6x – 5y = 0Add 5y on both sides to isolate 6x: 6x – 5y + 5y = 0 + 5y6x = 5yDivide both sides by 6 to solve for x: 6x/6 = 5y/6x = 5/6 y

Substitute the value of x into your base equation.x – y = 1Substitute for x: (5/6)y – y = 1

Solve for y.(5/6)y – y = 1Convert 1y into a fraction: (5/6)y – (6/6)y = 1(-1/6)y = 1Multiply both sides by the reciprocal of -1/6: (-6/1)(-1/6)y = 1(-6/1)y = -6

Substitute the value of y into your base equation and solve for x.x – y = 1x – (-6) = 1Add -6 to both sides to isolate x: x – (-6) +(-6) = 1 + (-6)x = -5

Substitute the values of x and y into both equations to double check the results. If the equations work out, then you have correctly solved your linear system.This result for this example is (-5, -6). Always write the answer in point form (x, y).

Graph the results by creating table with ordered pairs. Solve both equations for x or y. Which variable you choose is up to you. For example, using the equation from Step 4, substitute 0 in place of y.x = 5/6 yx = (5/6) * 0x = New ordered pair = (0, 0)Repeat this step until you have at least three points to graph for each equation.

Graph the points and connect the dots for each line. The result in Step 8 should be the only place the two lines intersect.

About the Author

C.D. Crowder has been a freelance writer on a variety of topics including but not limited to technology, education, music, relationships and pets since 2008. Crowder holds an A.A.S degree in networking and one in software development and continues to develop programs and websites in addition to writing.