# Solving Systems of Equations by Elimination Worksheet, Examples, and Steps

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### Key Points about Solving Systems with Elimination

- The elimination method is a systematic approach that involves adding or subtracting equations to eliminate one of the variables.
- To solve systems of equations by elimination, one needs to follow a series of steps, including identifying the variable to eliminate, ensuring that the coefficients of the variable are the same in both equations, adding or subtracting the equations to eliminate the variable, and solving for the remaining variable.
- Solving systems of equations by elimination can be challenging, but with practice, it becomes easier.

## System of Equations Elimination Method: The Complete Guide

**Solving Systems of Equations by Elimination** is a method to solve a system of two linear equations. **Solving Systems of Equations by Elimination** follows a specific process in order to simplify the solutions. The first thing you must do when **Solving Systems of Equations by Elimination** is to multiply either equation so that when you add them vertically, one of the variables will cancel out. Next, you must add the equations together vertically to cancel out one variable. Then you solve for the variable that was not cancelled out. Use the solution of the variable to substitute back in to either original equation. The last step when **Solving Systems of Equations by Elimination** is to solve for the last variable by following order of operations.

Solving systems of equations by elimination is a fundamental concept in algebra. It involves using the elimination method to find the values of two or more variables in a system of equations. The elimination method is a systematic approach that involves adding or subtracting equations to eliminate one of the variables.

To solve systems of equations by elimination, one needs to follow a series of steps. First, one needs to identify the variable to eliminate and then ensure that the coefficients of the variable are the same in both equations. Next, one needs to add or subtract the equations to eliminate the variable. Finally, one can solve for the remaining variable and substitute the value back into one of the original equations to find the value of the eliminated variable.

Solving systems of equations by elimination can be challenging for some students, but with practice, it becomes easier. The elimination method is a powerful tool that can be used to solve a wide range of problems. Understanding the steps involved and practicing with examples is essential to mastering this concept.

**Common Core Standard: **8.EE.C**Related Topics: **Identifying One, None, Infinite Solutions, Solving Systems by Graphing, Solving Systems by Substitution**Return To: **Home, 8th Grade

## What is the Elimination Method?

The elimination method is a technique for solving systems of linear equations. It involves adding or subtracting the equations in the system to eliminate one of the variables. The goal is to create a new equation that only has one variable, which can then be solved for. Once one variable is solved for, the solution can be substituted back into one of the original equations to solve for the other variable.

The elimination method is particularly useful when the coefficients of one of the variables in the system are additive inverses. In other words, one coefficient is the negative of the other coefficient. This allows the two equations to be added together in a way that eliminates one of the variables.

To use the elimination method, the system of equations must be written in standard form, where the variables are on the left side of the equation and the constants are on the right side. The equations must also be lined up so that the variables are in the same order in each equation.

The elimination method can be used to solve systems of equations with two or more variables. However, as the number of variables increases, the process becomes more complicated and time-consuming. In some cases, it may be more efficient to use other methods, such as substitution or graphing.

## How to Solve Systems of Equations by Elimination

### Adding and Subtracting Equations

The elimination method is a technique used to solve systems of linear equations. This method involves adding or subtracting the equations in the system to eliminate one of the variables. To do this, the coefficients of one of the variables in each equation must be the same or opposite.

For example, consider the system of equations:

```
2x + 3y = 8
4x - 5y = 2
```

To eliminate the variable `y`

, we can multiply the first equation by `-5`

and the second equation by `3`

. This gives us:

```
-10x - 15y = -40
12x - 15y = 6
```

Now, if we add these two equations, we can eliminate the variable `y`

:

```
2x = -34
```

### Simplifying Coefficients and Constants

After eliminating one of the variables, the next step is to simplify the coefficients and constants to solve for the remaining variable. In the example above, we have:

```
2x = -34
```

To solve for `x`

, we can divide both sides by `2`

:

```
x = -17
```

Now that we have solved for `x`

, we can substitute this value back into one of the original equations to solve for `y`

. Let’s use the first equation:

```
2x + 3y = 8
```

Substituting `-17`

for `x`

, we get:

```
2(-17) + 3y = 8
```

Simplifying, we get:

```
-34 + 3y = 8
```

Adding `34`

to both sides, we get:

```
3y = 42
```

Dividing both sides by `3`

, we get:

```
y = 14
```

Therefore, the solution to the system of equations is `x = -17`

and `y = 14`

.

### Solving Linear Equations

It is important to note that the elimination method can only be used to solve systems of linear equations. Non-linear equations, such as quadratic or exponential equations, require different techniques to solve. Additionally, it is important to check the solution to ensure that it satisfies both equations in the system.

In summary, the elimination method involves adding or subtracting equations to eliminate one of the variables, simplifying the coefficients and constants, and solving for the remaining variable. This method can only be used to solve systems of linear equations.

## Elimination Method Steps

The elimination method is a technique used to solve systems of linear equations. The method involves eliminating one of the variables in the system by adding or subtracting the two equations. Once one variable is eliminated, the remaining variable can be solved for. Here are the steps involved in using the elimination method:

**Identify the variables**: The first step in using the elimination method is to identify the variables in the system of equations. Typically, the variables are represented by x and y.**Determine which variable to eliminate**: The next step is to determine which variable to eliminate. This is done by looking at the coefficients of the variables in each equation. The goal is to choose the variable that has the same coefficient in both equations, but with opposite signs.**Multiply one or both equations**: If the coefficients of the variables are not the same, you will need to multiply one or both equations by a constant to make the coefficients match. The goal is to make the coefficients of the chosen variable equal but with opposite signs.**Add or subtract the equations**: Once the coefficients of the chosen variable are equal but with opposite signs, add or subtract the two equations to eliminate the chosen variable.**Solve for the remaining variable**: After eliminating one variable, you can solve for the remaining variable by substituting the value of the eliminated variable into one of the original equations.**Find the solution**: Finally, you can find the solution to the system of equations by substituting the values of the variables into one of the original equations and solving for the remaining variable.

Using the elimination method can result in one or more solutions to the system of equations. The solutions represent the points where the two equations intersect on a graph.

## 3 Simple Solving Systems of Equations by Elimination Examples

Elimination is a powerful method for solving systems of equations. It involves adding or subtracting the equations in a way that eliminates one of the variables. This leaves us with a new equation that only involves the remaining variable, which we can solve for. Once we have the value of one variable, we can substitute it back into one of the original equations to solve for the other variable.

- Multiple one, or both, equations so that when you add them together one variable will cancel.
- After you have multiplied, add the equations together.
- Solve for the variable that was left over after the other one was cancelled out.
- Substitute your solution for the first variable back into either equation.
- Solve for the second variable.
- Your answer must be in an x and y coordinate.

Here are a few examples of solving systems of equations by elimination:

#### Example 1

Solve the following system of equations by elimination:

```
2x + 3y = 5
4x - 5y = -13
```

To eliminate the `x`

variable, we need to multiply the first equation by `-2`

and add it to the second equation. This gives us:

```
-4x - 6y = -10
4x - 5y = -13
```

Now, we can add the two equations together to eliminate the `x`

variable:

```
-11y = -23
```

Solving for `y`

, we get:

```
y = 23/11
```

Substituting this value back into one of the original equations, we can solve for `x`

:

```
2x + 3(23/11) = 5
2x = 2/11
x = 1/11
```

Therefore, the solution to the system of equations is `(1/11, 23/11)`

.

#### Example 2

Solve the following system of equations by elimination:

```
3x - 2y = 7
2x + y = 1
```

To eliminate the `y`

variable, we need to multiply the second equation by `-2`

and add it to the first equation. This gives us:

```
-4x - 2y = -2
3x - 2y = 7
```

Now, we can add the two equations together to eliminate the `y`

variable:

```
-x = 5
```

Solving for `x`

, we get:

```
x = -5
```

Substituting this value back into one of the original equations, we can solve for `y`

:

```
3(-5) - 2y = 7
y = -4
```

Therefore, the solution to the system of equations is `(-5, -4)`

.

#### Example 3

Solve the following system of equations by elimination:

```
2x + y = 10
3x - 4y = 5
```

To eliminate the `y`

variable, we need to multiply the first equation by `4`

and add it to the second equation. This gives us:

```
8x + 4y = 40
3x - 4y = 5
```

Now, we can add the two equations together to eliminate the `y`

variable:

```
11x = 45
```

Solving for `x`

, we get:

```
x = 45/11
```

Substituting this value back into one of the original equations, we can solve for `y`

:

```
2(45/11) + y = 10
y = -13/11
```

Therefore, the solution to the system of equations is `(45/11, -13/11)`

.

These examples demonstrate how elimination can be used to solve systems of equations. By carefully manipulating the equations, we can eliminate one of the variables and solve for the other. With practice, this method can be used to solve more complex systems of equations as well.

## 5 Quick Elimination Method Problems

## Solving Systems by Elimination FAQ

### What is the elimination method for solving systems of equations?

The elimination method is a technique used to solve a system of linear equations. It involves adding or subtracting equations to eliminate one of the variables, resulting in a new equation with only one variable. This process is repeated until all variables are isolated and their values can be determined.

### How do you use the elimination method to solve systems of equations?

To use the elimination method, you need to identify which variable to eliminate first. You then add or subtract the equations to eliminate that variable. Once you have eliminated one of the variables, you can solve for the other variable. You then substitute the value of that variable back into one of the original equations to solve for the other variable.

### What are the steps for solving systems of equations by elimination?

The steps for solving systems of equations by elimination are as follows:

- Identify which variable to eliminate first.
- Multiply one or both equations by a constant to create opposite coefficients for the variable you want to eliminate.
- Add or subtract the equations to eliminate the variable.
- Solve for the remaining variable.
- Substitute the value of the remaining variable back into one of the original equations to solve for the other variable.

### Can you provide an example of solving systems of equations using elimination?

Sure. Consider the following system of equations:

```
2x + 3y = 5
4x - 2y = 10
```

To solve this system using the elimination method, we can eliminate the variable `y`

by multiplying the first equation by `-2`

and adding it to the second equation:

```
-4x - 6y = -10
4x - 2y = 10
```

This results in the equation `-8y = 0`

, which we can solve for `y`

to get `y = 0`

. We can then substitute this value back into one of the original equations to solve for `x`

, which gives us `x = 5/2`

.

### What are some common mistakes to avoid when using the elimination method?

One common mistake is forgetting to multiply the equations by a constant to create opposite coefficients for the variable you want to eliminate. Another mistake is making a sign error when adding or subtracting the equations. It is also important to double-check your solution by substituting the values back into the original equations to ensure they satisfy both equations.

### How does the elimination method compare to the substitution method for solving systems of equations?

The elimination method and the substitution method are both techniques used to solve systems of linear equations. The elimination method involves adding or subtracting equations to eliminate a variable, while the substitution method involves solving one equation for one variable and substituting that expression into the other equation. The elimination method is typically faster when dealing with larger systems of equations, while the substitution method can be easier to use when one of the equations is already solved for one variable.

### Why does the elimination method work when solving a system of equations?

The elimination method works because adding or subtracting equations does not change the solution to the system of equations. By eliminating one of the variables, we are left with a new equation with only one variable, which we can solve to find the value of that variable. We can then substitute that value back into one of the original equations to solve for the other variable.

## Elimination Method Worksheet: Video Explanation

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

**Watch the free Solving Systems of Equations by Elimination video on YouTube here: Solving Systems of Equations by Elimination**

**Video Transcript:**

This video on solving systems of equations by elimination. You can get the solving systems of equations worksheet used in this video for free by clicking on the link in the description below.

Here’s number one on our solving systems of equations by elimination worksheet. This problem gives us two separate equations in our system. The first equation is 3x minus y equals 7 and the second equation is negative 2x plus y equals 2. When you do the elimination method to solve systems of equations you’re going to add each part of your equation vertically so that one or more of the terms will cancel.

In the case of solving systems by using the elimination method you have to visualize how you couldn’t add these vertically to see what terms or variables will cancel. Now it does not matter if you cancel out the X or the Y but at least one of them must cancel.

We can look at our equations and we can see that if we were to add these straight down the Y’s would cancel so what would happen is you would do 3x plus negative 2x would be 1x or 3x minus 2x would be 1x and then negative y plus 1y. This is negative 1 y plus 1 Y would be 0 Y and then 7 plus 2 is 9. In other words what has happened is we have taken Y because we have 0y now and we have canceled it out. It is now gone and we are left with 1x or just x equals 9. Now we know that x equals 9 we can take x equals 9 and we can substitute x equals 9 back into one of our equations. The elimination method is just one way to figure out how to solve a system of equations but this is the easiest way to learn how to do elimination.

n this case we’re going to take this 9 and we’re going to substitute it in back into our first equation. This 9 because X is equal to 9 can get substituted in for X. Now we have 3 times 9 which is what we substituted in 4 minus y equals 7. So now we know X is equal to nine but we have to find y because we have to know the X and the y coordinate for this solution of this system we took our X which was nine and we substituted in for X then we’re going to simplify this. 3 times 9 is 27 minus y equals 7. Then we have to solve for y.

We’re going to go ahead and subtract 27 from this side subtract 27 from this side these cancel you have negative y equals 7 minus 27 is negative 20 and then we have to get rid of this negative. We divide both sides by negative 1, these cancel and you have positive y equals positive 20. We know our x coordinate and we know our Y is now 20. We know that our solution is the coordinate 9, 20 and that’s going to be our answer.

The next problem on our solving system of equations by elimination worksheet is number two. We have to first know how to solve by elimination. X plus y equals 4 and then 2x minus 3y equals 18. When we go to eliminate these you will notice that if we were to add these straight up or down nothing would cancel. If we added these right now for instance X plus 2 X would be 3 X and then Y minus 3 y would be negative 2y and then 4 plus 18 would be 22. Nothing cancelled so that means we can’t do this, we have to multiply one or both equations so that we get a situation where when we add vertically something will cancel.

Our first equation can be multiplied that when we add these they will cancel now the easiest thing to multiply the top equation by would be 3 because if we multiplied everything by 3 we would get 3y and then the Y’s would cancel. The positive 3y and the negative 3y would cancel when we add we could also multiply by negative two because if we multiply by negative two we could add the x’s and that negative two and the positive 2x would cancel. Tou can pick and choose if you want to do X or if you want to cancel Y but in this case I’m going to cancel Y.

What we’re going to do is we’re going to go ahead and multiply the top equation by three so that we will get positive 3y and then this times three we’re multiplying everything times three. We have to distribute this three to both the X and the y on this side. We’re going to say 3 times X is 3x and then three times y is 3y and then is equal to 4 times 3 which is 12. Then our second equation we don’t have to change because we know that this positive 3y and this negative 3y are going to cancel. We’re just going to rewrite the second equation over here as 2x minus 3y equals 18.

Then what we’re going to do is we’re going to add vertically. We will add both equations together because we know that the Y’s are going to cancel. 3x plus 2x is 5x and then 3y plus negative three Y would be 0y or the Y’s would just cancel. And then 12 plus 18 is 30 now the 0y has gone. We’ll just cancel it like that. We’ve got 5x equals 30 now. Then in order to solve for X we’re going to divide both sides by 5. These cancel and then 30 divided by 5 is 6.

Now we know x equals 6 we’re going to take x equals 6 and we’re going to substitute it back into either equation. In the case of this system it is easiest to substitute the 6 in or the first equation because our first equation is just X plus y equals 4 and this does not have a coefficient. We will rewrite our equation. We have X which is 6 so we substituted six in for X plus y equals four. Then we have to solve for y. We’ll subtract six from both sides, these cancel and we get y equals negative two. Our solution is x equals six y equals negative two and that’s the answer. Try the practice problems by downloading the free solving system of equations by elimination worksheet above.

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