How to Solve a Set of Linear Equations
Linear equations are mathematical expressions that contain variables and numerical constants. A linear expression can be defined as one in which the variables are not raised to any power and not multiplied by one another. A set of linear expressions is just a series of these equations which all contain the same variables. Solving a set means finding number values for the variables that will make all the equations true. We can always solve a set of linear equations as long as we have as many different equations as we have variables. The simplest way to solve these sets is by substitution.
Other People Are Reading
Instructions
-
Solving a Set of Linear Equations by Substitution
-
1
Examine the set of linear equations and ensure that you have as many distinct equations as you have variables. For example, in the set consisting of the equations 2x + 5y = 10 and 2x - 2y = 24, there are two equations and two variables, so this set can be solved.
-
2
Rearrange one of the equations so a single variable with no multiplying numerical value is alone on one side of the equals sign. In our example, we would rearrange the first equation to read x = (10 - 5y)/2, which can be simplified further to x = 5 - (5/2)y.
-
-
3
Take the portion of the equation you have just rearranged which is on the opposite side of the equals sign from the lone variable. Substitute that portion wherever you see that same variable in the other equation. So we would substitute 5 - (5/2)y wherever we see x in our second equation. The second equation then becomes 2[5 - (5/2)y] - 2y = 24 which simplifies to 10 - 5y - 2y = 24.
-
4
Solve the second equation for the single variable that it now contains. The second equation in our example would become -7y = 14, and solving would give y = -14/7 or y = -2.
-
5
Substitute the numerical value just found for the second variable back into the first equation. Our first equation would become x = 5 - (5/2)(-2).
-
6
Solve for the remaining variable in the first equation. In the example, the first equation now simplifies to x = 5 + 5 so x = 10. We have now solved the set of equations by finding values for x and y that will make both equations true.
-
1
Tips & Warnings
You can always double check your answer by taking your calculated values of the variables, placing those values in the original equations and making sure the equations are now true.
This exact same technique can be used to solve sets with three or more variables as well, just with a few more substitution steps.
Remember to maintain equality on either side of the equals sign when rearranging equations. Whatever mathematical operation is done to one side must be done to the other.
References
Resources
- Photo Credit Phil Ashley/Lifesize/Getty Images
