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# 3 Steps to Successfully Solve Systems of Linear Equations

Get the free Systems of Linear Equations worksheet and other resources for teaching & understanding solving Systems of Linear Equations

## Here’s How to Solve Systems of Linear Equations

There are three different types of solutions to Systems of Linear Equations. There can be One Solution, No Solution, and Infinite Solutions. When Identifying One, None, and Infinite Solutions to equations you must first solve the equation for the variable. When solving for the variable, you need to follow the rules and properties for solving equations. Once the equation has been solved, there can be the One Solution, No Solution, or Infinite Solutions.

Identifying One, None, and Infinite Solutions to Systems of Linear Equations can be simple once the equation has been solved for the variable. For the equation to have One Solution, your answer will be a variable equals a number. For the equation to have No Solution, your answer will be an Untrue Statement of numbers. For the equation to have Infinite Solutions, your answer will be a True Statement of numbers.

Common Core Standard: 8.EE.C.8

## A Short Guide to complete any Systems of Linear Equations Example

How can you tell the difference between what is one solution, what is no solution, or what is infinite solutions? There are three unique answers for Systems of Linear Equations. There can be One Solution, No Solution, and Infinite Solutions. When Identifying One, None, and Infinite Solutions to systems you should solve for the variable. When solving for the variable, you have to use the rules for solving equations. When the equation has been solved, there can be the One Solution, No Solution, or Infinite Solutions.

Distinguishing One, None, and Infinite Solutions to Systems of Linear Equations can be straightforward once the system has been solved for the variable. For the system to have One Solution, your answer will be a variable equals a number. For the system to have No Solution, your answer will be an Untrue Statement of numbers. For the equation to have Infinite Solutions, your answer will be a True Statement of numbers.

## 4 Easy Steps for Solving Systems of Linear Equations Problems

1. Solve the System of Linear Equations for the variable.
2. If you get x equals a number as your answer, then there is one solution to the system.
3. If you get a number equals a different number, then there are no solutions to the system.
4. If you get a number equals the same number, then there are infinite solutions to the system.

## Solving Systems of Equations Practice Problems Quiz

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One, None, Infinite Solutions Quiz

Click Start to begin the practice quiz!

1 / 5

State whether each equation has One Solution, No Solution, or Infinite Solutions.

-5x + 32 - 4x = 30 - 9x

2 / 5

State whether each equation has One Solution, No Solution, or Infinite Solutions.

12x + 6 = 2x - 24

3 / 5

State whether each equation has One Solution, No Solution, or Infinite Solutions.

3x + 15 = 3x + 15

4 / 5

State whether each equation has One Solution, No Solution, or Infinite Solutions.

-3x - 9 = 5x - 9 - 8x

5 / 5

State whether each equation has One Solution, No Solution, or Infinite Solutions.

3x + 18 = 3(x - 6)

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## Watch the video where we complete our Systems of Linear Equations Worksheet

Watch our free video on how to solve Systems of Linear Equations. This video shows how to solve problems that are on our free Solving Systems of Linear Equations worksheet that you can get by submitting your email above.

Watch the free Systems of Linear Equations video on YouTube here: Systems of Linear Equations

Video Transcript:

This video is about systems of linear equations. You can get the one solution no solution infinite solutions worksheet we use in this video for free by clicking on the link in the description below.

Here we are the first problem for systems of linear equations. We need to state whether this system has one solution, no solutions, or an infinite amount of solutions. In order to determine what type of solution this system has, you have to know the difference between the three types.  If it has one solution, that means you’re going to have an answer where x equals a number. It would look like any typical answer that you get for an equation for a system. To have no solutions that would mean that you get an untrue statement. What this means is you would get some number, let’s say in this case 17, is equal to something that it is not, which would be let’s say 8. This is not true 17 does not equal 8. That means that the system would have no solutions. Then finally the last type would be an infinite amount of solutions, and that is when you get a true statement. This means that you could have a number that equals the same number. If we had 17 equals 17, that is true right, 17 doesn’t equal 17 so that means it would be an infinite amount of solutions.

Looking at our problem, we’re given 12 X plus 6 equals 2x minus 24. We have to solve this equation for X. When we go to solve this the first thing we need to do is we need to get all of the variables on one side and then we need to get all the constants on the other side. The first thing we’re going to do is we’re going to subtract 2x from both sides so that we get the X is only on one side. 12x minus 2x is 10x then you bring down your plus 6 and you bring down your minus 24. Then we have to get the constants on the opposite side of the variable. We will subtract 6 from both sides, these will cancel and you end up with and X is equal to negative 30. Then the last step is to divide both sides by 10. When we do this we will get x equals negative 3.

Our answer is x equals negative 3. That means this has one solution to the system of linear equations because we know that X is equal to negative 3 and only negative 3. Our answer can only be negative 3 which means it’s one solution. This is a system one solution example problem.

Moving on to the second problem for system of linear equations, we are given 3 x plus 15 equals 3x plus 15. Now the first step to solve this is we have to get the variables on one side together. What we’re going to do is we’re going to subtract this 3x and then we’re going to subtract 3x from this side. When we do this the 3 X’s here cancel and also the 3 X’s here cancel. We are left with 15 equals 15. We bring these down. Our X’s are cancelled and we’re left with just two constants.

In this case the constants are equal to each other. 15 equals 15. When you have two constants that are equal to each other that means that the answer for the system of linear equations is infinite solution example. That means no matter what you put in for X you will always get a true statement. You could put any number in for X and you will always get something that is true. Because it’s always true you can use any number and that means you have an infinite amount of answers that will be true or that will answer this system. This is a system of equations infinite solutions example problem.

The last problem we’re going to go over is number 4. Number 4 gives us the system 3x plus 18 equals 3 times the quantity X minus 6. The first step in this equation is to distribute the 3 to both the X and the negative 6. When we do this we will multiply 3 times X which is 3x and then 3 times negative 6 which is negative 18. The next step is to get all the constants on one side together. We’re going to subtract this 3x over here and we’re also going to subtract this 3x over here. When we do that these X’s cancel and these X’s also cancel. We’re going to bring down the 18 on this side and on this side we have negative 18. Our solution is 18 equals negative 18.

That is not true that is an untrue statement 18 does not equal negative 18. They are different numbers that means that this system has no solution. There’s no number that you can substitute in for X that will get you a true statement. That means there are no solutions to this. This is a system of equations no solutions example problem.Try all the practice problems by downloading the free one solution, no solution, infinite solutions worksheet above.

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