What Is The Elimination Method In Math
In math, the elimination method is used to solve a system of linear equations. It is the most widely used and simple method as it involves fewer calculations and steps. In this method, we eliminate one of the two variables and try to solve equations with one variable. The value found here can be substituted in any of the given equations to find the value of the other variable as well.
Summary Of The Methods For Solving Linear Systems
We have developed three methods for solving linear systems of two equations with two variables. In this section, we summarize the strengths and weaknesses of each method.
The graphing method is useful for understanding what a system of equations is and what the solutions must look like. When the equations of a system are graphed on the same set of axes, we can see that the solution is the point where the graphs intersect. The graphing is made easy when the equations are in slope-intercept form. For example,
The simultaneous solution corresponds to the point of intersection. One drawback of this method is that it is very inaccurate. When the coordinates of the solution are not integers, the method is practically unusable. If we have a choice, we typically avoid this method in favor of the more accurate algebraic techniques.
The substitution method, on the other hand, is a completely algebraic method. It requires you to solve for one of the variables and substitute the result into the other equation. The resulting equation has one variable for which you can solve. This method is particularly useful when there is a variable within the system with coefficient of 1. For example,
The elimination method is a completely algebraic method that makes use of the addition property of equations. We multiply one or both of the equations to obtain equivalent equations where one of the variables is eliminated if we add them together. For example,
Solving Systems Of Equations By Elimination With Multiplication
What happens if the coefficients of a variable arent additive inverses in a system of equations? For example, consider the system below.
If we add these equations as they are now, the y terms would become -2y+=-4y and the x terms would become 4x+7x=11x. We wont have eliminated any variables.
However, if we multiply one of the equations by the factor -1, we end up with coefficients that are additive inverses. Lets try it:
Now, we can add our equations to eliminate the y variable and solve for x.
To solve other systems of equations by elimination, well need to rewrite both equations to eliminate a variable. For example, consider the system below.
We could multiply the first equation by -2 and the second equation by 3 to allow us to eliminate x when we add the equations. Lets use this process to solve the system by elimination.
|1. Determine which variable will be eliminated. If necessary, rewrite one or both equations to make the coefficients of that variable additive inverses.||}=} -6x + 18y = -12|
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Why Is Elimination Method Better
The elimination method is considered better than other methods for solving a system of equations because it makes the calculations easier by eliminating a variable. Once a variable is eliminated, it becomes easier to determine the value of another variable and then use this value to find the value of the eliminated variable.
Solving System Of 3 Equations Using Elimination Method
To solve a system of three linear equations with the elimination method, first, make sure that the equations are written in the standard form Ax+By+C=0 or Ax+By=C without any fractional coefficient. Take any two equations as per your comfortability and select a variable to eliminate. Eliminate the chosen variable. Now, select another pair of equations out of the given three equations and eliminate the same variable. This way you will get two equations with only two variables. Solve those using the elimination method steps mentioned above and find the values of those 2 variables. Substitute the values in any of the given equations to find the value of the third variable.
Let’s solve three equations 3x-y+2z=5, 4x+2y-z=6, and 5x-3y+z=1 for a better understanding.
Now, we have found that x=1. Substitute this value in equation P, we get 9×-y=7.
Now put the values of x and y in the third equation 5x-3y+z=1, and we get z=2. Therefore, x=1, y=2, and z=2.
Important Notes on Elimination Method
- The elimination method is used to solve a system of equations.
- This method is easy and makes the calculations easier as it eliminates one variable and hence, reduces the calculations.
- We make the coefficient of a variable identical to eliminate the corresponding variable.
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What Are The Steps Involved In The Elimination Method
The steps involved in the elimination method are given below:
- Choose any one variable to eliminate. Multiply or divide both the equations with a non-zero constant to make the coefficients of that variable equal.
- Add or subtract the resultant equations such that the chosen variable gets eliminated.
- Simplify and find the value of the other variable.
- Substitute that value in any of the given equations to find the value of the eliminated variable.
Solving Systems Of Equations By Substitution Method
The steps to apply or use the substitution method to solve a system of equations are given below:
- Step 1: Simplify the given equation by expanding the parenthesis if needed.
- Step 2: Solve any one of the equations for any one of the variables. You can use any variable based on the ease of calculation.
- Step 3: Substitute the obtained value of x or y in the other equation.
- Step 4: Now, simplify the new equation obtained using arithmetic operations and solve the equation for one variable.
- Step 5: Now, substitute the value of the variable from Step 4 in any of the given equations to solve for the other variable.
Here is an example of solving system of equations by using substitution method: 2x+3=0 and x+4y+2=0.
Step 1: So, here we can simplify the first equation to get 2x + 3y + 15 = 0. Now we have two equations as,
2x + 3y + 15 = 0 _____
x + 4y + 2 = 0 ______
Step 2: We are solving equation for x. So, we get x = -4y – 2.
Step 3: Substitute the obtained value of x in the equation . i.e., we are substituting x = -4y-2 in the equation 2x + 3y + 15 = 0, we get, 2 + 3y + 15 = 0.
Step 4: Now, simplify the new equation. We get, -8y-4+3y+15=0
-5y + 11 = 0
-5y = -11
y = 11/5
Step 5: Now, substitute the value of y in any of the given equations. Let us substitute the value of y in equation .
x + 4y + 2 = 0
x + 4 × + 2 = 0
x + 44/5 + 2 = 0
x + 54/5 = 0
x = -54/5
Therefore, after solving the given system of equations by substitution method, we get x = -54/5 and y= 11/5.
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State The Concept Of Elimination Method
In the elimination method you either add or subtract the equations to get an equation in one variable.
When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.
What Does Elimination Mean In Algebra
The elimination method is where you actually eliminate one of the variables by adding the two equations. In this way, you eliminate one variable so you can solve for the other variable. In a two-equation system, since you have two variables, eliminating one makes the process of solving for the other quite easy.
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Solving Systems Of Equations By Elimination Examples
Lets take a closer look at the system below.
First, notice that the coefficients for the y terms are 5 and -5. These two numbers are additive inverses, which means they have a sum of 0. Therefore, if we were to add these equations, the y terms will cancel each other out.
But is adding equations a legitimate means to solve a system of equations? Consider the Addition Property of Equality, which states:
If a=b, then a+c=b+c.
Were going to use this property to solve systems of linear equations by elimination.
To start, lets think about the second equation in two parts. We could separate each part and rewrite them as follows:
We know these statements are true because 6x-5y=3.5. Therefore if one side of this equation equals z, then the other equals z as well.
Now, we can use the Addition Property of Equality to rewrite the first equation in our system, 2x+5y=12.5, as:
and then substitute in our values of z to get:
Weve just added our two equations! When we simplify our new equation, well eliminate the y term so that we can solve for x.
Next, we can substitute 2 for x in one of our original equations and solve for y.
So, the solution to our system is . We can check our work by graphing, as shown below.
Elimination Method: No Solutions
As we know that equations of two parallel lines have no solutions. So, if we solve any such equations using the elimination method we get the answer as two unequal numbers on both sides of the unequal sign. For example, 08, -20, etc. In such cases, we cannot eliminate only one variable. Both the variables get eliminated. For example, let us solve two equations 2x – y = 4 and 4x – 2y = 7 by the elimination method. In order to make the x coefficients equal in both the equations, we multiply equation by 2 and equation by 1. By doing so we get, 4x – 2y = 8 and 4x – 2y = 7 . Now, if we try to subtract equation 4 from equation 3, we get, 0=1 as both the variables are getting eliminated. There is no other way to solve these equations as the solutions are inconsistent. So, in the elimination method when there is no solution, we get the result in this form.
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Solving Systems Of Equations In Two Variables By The Addition Method
A third method of solving systems of linear equations is the addition method, this method is also called the elimination method. In this method, we add two terms with the same variable, but opposite coefficients, so that the sum is zero. Of course, not all systems are set up with the two terms of one variable having opposite coefficients. Often we must adjust one or both of the equations by multiplication so that one variable will be eliminated by addition.
How To: Given A System Of Two Equations In Two Variables Solve Using The Substitution Method
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How Do You Solve An Equation With Two Variables
Divide both sides of the equation to solve for x. Once you have the x term on one side of the equation, divide both sides of the equation to get the variable alone. For example: 4x = 8 2y. /4 =
How is the elimination method used in Algebra?
The elimination method is where you actually eliminate one of the variables by adding the two equations. In this way, you eliminate one variable so you can solve for the other variable. In a two-equation system, since you have two variables, eliminating one makes the process of solving for the other quite easy. Lets try one:
Solve A System Of Equations By Elimination
The Elimination Method is based on the Addition Property of Equality. The Addition Property of Equality says that when you add the same quantity to both sides of an equation, you still have equality. We will extend the Addition Property of Equality to say that when you add equal quantities to both sides of an equation, the results are equal.
For any expressions a, b, c, and d,
To solve a system of equations by elimination, we start with both equations in standard form. Then we decide which variable will be easiest to eliminate. How do we decide? We want to have the coefficients of one variable be opposites, so that we can add the equations together and eliminate that variable.
Notice how that works when we add these two equations together:
Lets try another one:
This time we dont see a variable that can be immediately eliminated if we add the equations.
But if we multiply the first equation by 2, we will make the coefficients of x opposites. We must multiply every term on both sides of the equation by 2.
Now we see that the coefficients of the x terms are opposites, so x will be eliminated when we add these two equations.
Add the equations yourselfthe result should be 3y = 6. And that looks easy to solve, doesnt it? Here is what it would look like.
Well do one more:
We can make the coefficients of x be opposites if we multiply the first equation by 3 and the second by 4, so we get 12x and 12x.
This gives us these two new equations:
When we add these equations,
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Elimination Method For System Of 3 Equations
To use the elimination method to solve a system of three linear equations, first ensure that the equations are written in the standard form Ax+By+C=0 or Ax+By=C without any fractional coefficients.
3x + 4y + 2z = 9
4x 4y + 3z = 3.
2x 4y + 3z = 1
Add equation and then we get
5x + 5z = 10
x + z = \\)
Now add equation and
In equation , substitute y = 0.
2x + 0 = 3
x = \frac
As a result x = \ and y = 0
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How To Solve Simultaneous Equations Using Elimination Method
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Have you ever had a simultaneous problem equation you needed to solve? When you use the elimination method, you can achieve a desired result in a very short time. This article can explain how to perform to achieve the solution for both variables.
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How To Do Gaussian Elimination
Now we are going to see with a solved example how to solve a system of linear equations using the Gaussian elimination method:
First of all, we find the augmented matrix of the system of equations:
As we will see later, it is better if the first number of the first row is 1. Therefore, we are going to change the order of rows 1 and 2:
The goal of the Gaussian elimination method is to make the numbers below the main diagonal 0. So, we have to transform the red numbers to 0:
To eliminate these numbers we must do the appropriate elementary rows operations.
For example, the number -1, which is the first element in the second row, is the negative of 1, the first element in the first row. So if we add the first row to the second row, the -1 will be eliminated:
In this way we have transformed the -1 into a 0.
Now we are going to zero the number 2 out. To do so, we add the first row multiplied by -2 to the third row:
And the following matrix is obtained:
Now we have to convert the -8 to 0. To do this, we multiply the third row by 3 and add the second row multiplied by 8:
Therefore, we obtain the following matrix:
As you can see, with all these transformations we have made all the numbers below the main diagonal 0. That is, the matrix is in row echelon form .
And finally, to solve the system we have to find the values of the unknowns from the bottom up. Since the last equation only has one unknown, and therefore, we can easily find its value:
More Complex Systems Of Equations With Multiplication
Finally, well encounter systems where we need to rewrite equations more extensively to eliminate a variable. For example, consider the system below.
Before we add these equations, we need to move the y term to the opposite side of the equals sign in the first equation and multiply the second equation by -2.
3x – 4y =4y-2 – 4y \rightarrow 3x – 4y = -2
-2=-2\rightarrow -10x + 4y = -12
Now, we can eliminate y terms and solve for x.
To summarize, the key to effectively rewriting the equations is selecting the correct factor to multiply one or both equations to create additive inverses. Additive inverses will cancel each other out and thus eliminate that variable when the equations are added.