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Methods of solving systems of equations

substitution method

A systems of equations is a set of multiple equations that need to be solved. The goal is to find the unknown numbers. The number of equations equals the number of variables in the equations.

For example, we need a system of two equations to solve equations that have two variables. There are 4 methods of solving systems of equations:

Substitution method

Substitution is a simple method in which we solve one of the equations for one variable and then substitute that variable into the other equation and solve it. For example, if we want to solve the first equation for variable X we need to rewrite the equation so it has the form of X = something. Then we need to substitute the X in the second equation with that “something” that we got in the first equation and then solve it. It is a lot easier if we show this method on a couple of examples.

substitution method step one

To solve the system of two equations above we need to follow these steps:

● First we need to solve one of the equations for one of the variables. Let’s choose the first equation and solve it for variable X. That means that we need to rewrite the equation in form of \ x = something. First we are going to transfer 4y to the other side of the equation. The sign of 4y will change into minus. It should look like this:

first step substitution method

We now need to divide both sides of the equation by 2 to get \ x = something. The result is:

second step substitution method

● In this step we are going to substitute the X in the second equation with the result for X from the first equation.

third step substitution method

That means that instead of X in the second equation we are going to put \ 3 - 2y and solve the equation for y. The resulting equation will only contain one variable and it can be solved.

forth step substitution

We need to get rid of the parentheses so we are going to multiply the numbers in the parentheses with 3. Then we are going to transfer variables on one side and everything else on the other side of the equation.

procedure system of equations substitution

● The result for y is \ y = 1. To find X we need to substitute y in the equation we got in the first step ( x = something) with the result we got in the previous step, in our example 1.

procedure solving system of equation substitution second step

The system of equations is solved. The results are \ x = 1 and \ y = 1.

● To check if the results are right we can choose one of the equations and substitute the values of x and y and check if the results on both sides of the equation are equal. We will choose the first equation:

verification solution

The results on both sides are equal. That means that the results for x and y are right.

Now you understand how to solve systems of equations by using substitution method but if we know all methods we will be able to solve every task with the most appropriate method.

system of equations methods

Let’s try to solve this system of equations using all methods.

Graphing method

graphing linear equation

Addition method

addition method

Gaussian elimination method

elimination method

And well known substitution method that we mention earlier:

We can write \ y = 5 - x. Now substitute this value of in the y second equation.

substitution method

Therefore, the solutions of all systems are (3, 2).

See few more example so you can get a clearer picture before you start with worksheets.

\ 3x + y = 9
\ 2x + 2y = 10

● Now, we are going to start with equation two, and solve it for y. First we are going to transfer 2x to the other side of the equation, and the divide both sides by 2.

\ 2x + 2y = 10
\ 2y = 10 - 2x
\ y = 5x

● To solve the first equation we need to substitute y in the first equation with the result for y in the previous step. That means that we need to put (5 x) in the first equation instead of y. Note that we always write the result in parentheses. That should always be done to avoid mistakes that can lead to false results.

\ 3x + (5 - x) = 9
\ 3x + 5 - x = 9
\ 3x - x = 9 - 5
\ 2x = 4
\ x = 2

● Now that we have the result for x, to find y, we only need to substitute x with value of 5 in the equation.

\ y = 5 - x
\ y = 5 - 2
\ y = 3

The results of the system of two equations is \ x = 2 and \ y = 3.

● To check if the results are right lets put our results for x and y in the first equation:

\ 2x + 2y = 10
\ 2 * 2 + 2 * 3 = 10
\ 4 + 6 = 10
\ 10 = 10
We got the right results.

Systems of equations worksheets

  Graphing - simple (6.3 MiB, 602 hits)

  Graphing - advanced (6.4 MiB, 631 hits)

  Substitution - simple (148.3 KiB, 759 hits)

  Substitution - advanced (143.0 KiB, 393 hits)

  Systems of equations - Elimination - Simple (344.3 KiB, 124 hits)

  Systems of equations - Elimination - Advanced (380.8 KiB, 128 hits)

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