Solving Systems of Linear Equations

Systems of linear equations can be solved in four unique ways. You can use Substitution, which is an algebraic form of solving the systems. You can also use linear combinations, which is a method that involves adding multiples of the given equations. Another method that can be used to solve systems of linear equations is graphing, a solution which shows all solutions of a system. The last method that you can use to solve systems of equations is matrices. To solve systems of equations using this method, you insert the coefficients respectively into a matrix. All these systems have their advantages and disadvantages.

Substitution is a common method of solving systems of linear equations. This method works when one of the equations given has a lone variable on one side of the equation. That equation must be substituted into the other equation, combining to make one equation. Here is an example:

1. 4x + 3y = 12

2. Y = 2x + 5

The second equation must now be substituted for y in equation 1, because y is the separate variable.

Now you are left with: 4x + 3(2x+5) = 12

Now solve the equation: 4x + 6x +15 = 12

Subtract 15 from 12 to get it on the right side. 10x = -3

Now simply solve for x.

X= -3/10

Now do the same thing to find y.

Substitution is most useful when one of the two given equations already has an isolated variable, like equation 2. Substitution will give you an exact answer, unlike graphing. It is a method easily carried out, but limited to systems of linear equations that contain an equation with an isolated variable so if neither of the equations that you are provided with contain an isolated variable, you must change them into y= or x= etc. form.

Linear combination is a fairly easy way to solve systems of linear equations. It involves eliminating a variable in order to make the system more easily solved. Linear combination would not be useful if an equation was given that already contained an isolated variable, it could be done, but substitution would be more sensible to use. Here is an example:

First, you must obtain coefficients for a variable that differ only in sign, so that they may be cancelled out to simplify the equations. One way to do this is multiplying an equation or both equations by the right number.

1. 4x + 4y = 6 You will multiply this equation by 3

2. -2x – 3y = -1 You will multiply this equation by 4


1. 12x +12y = 18

2. -8x – 12y = -4

Now you can add both equations together, and the y variable will be eliminated because its coefficients differ only in sign. Then solve for the last remaining variable.

1. 4x = 14

2. X = 3.5

Linear combination is useful because it allows for canceling or eliminating one of the variables in the equations for a simpler solution. This is done simply by multiplying one or both of the equations by numbers, in hopes to make the coefficients the same number with an opposite sign. Then they cancel each other out, and can be added together and solved.

Another process to solve system of equations is graphing. Graphing is great because it gives a visual diagram of the system. However it is not entirely accurate when done by hand, and can be hard to read. To solve using graphing, two aspects are required: the slope, and the y intercept. To solve a linear system using graphing, you plot the y intercept of each equation you are provided with, on a coordinate graph. Then use the slope (rise over run) to find the other point. Then draw a line between the two points and extend it outward until you locate the point of intersection. Then you can find the ordered pair of that point. Graphing is a good method because it visually displays the equations, however it is much more inaccurate then other methods.

The final method for solving is Matrices. Matrices are a much easier way to solve systems that contain three or more variables. Usually, Linear Combination could be used, but since it is a monotonous process, matrices may be the best method. Since matrices are unable to be divided, a matrix must be cancelled out by multiplication of its inverse. The inverse (when multiplied) will equal the identity matrix, which is similar to multiplying by “1”. The coefficients in a system must be inserted into a matrix in order. Here is an example.

-2x – 3y = -26

3x + 4y = 36

Matrix A ([A]) will be:

[-2 -3]

[3 4]

Matrix B ([B]) will be:



The equation should look like this:

[-2 -3] * [x] = [-26]

[3 4] [y] [36]

Then multiply [A]-1 * [B]

[-2 -3] -1 * [-26]

[3 4] [36]

Remember, Matrix multiplication is not commutative, so [B] * [A]-1 will give you a different answer than [A]-1 * [B]. One disadvantage of using matrices to solve systems of linear equations, is the time consuming process. However it is a good method for complex problems with many variables.

Systems of Linear Equations can be solved in numerous ways. Substitution is a nice concise algebraic method to solve systems that have an equation with an isolated variable, such as x=7y – 9. Substitution is very accurate, unlike graphing; however it is limited to equations that contain one lone variable, so if your equations do not contain one, you must make them into y= or x= etc. form to isolate a variable. Linear Combination is another method that is possible to use to solve systems. Linear combination is a good method because it enables canceling or eliminating of one of the variables in the equations for a simpler solution. Graphing is a desirable method to use for solving systems because it visually displays the solution. The disadvantage of graphing is the inaccuracy of it compared to other options of methods. Using Matrices is a long tedious process but very accurate and useful on large systems. Systems of Linear equations have many options with which to be solved.

Source by Paul Kennard

Leave a Reply