Solution of linear equations

!±8± Solution of linear equations

Systems of linear equations can be solved in four different ways. You can replace, which is a form of the solution of algebraic systems. You can also linear combinations, which includes a method to add multiple dates of equations. Another method that can be used to solve systems of linear equations is a graphical representation of a solution that has all the solutions of the system. The last method can be used to solve systems of equations, matrices. For solving systems of equationsWith this method, you add each of the coefficients in a matrix. All these systems have their advantages and disadvantages.

The replacement is a common method for solving linear equations. This method works if one of the equations was a variable on only one side of the equation. This equation must be replaced in the equation other, combine to make an equation. Here's an example:

1 4x + 3y = 12

Y 2 = 2x + 5

The second equation mustnow be replaced in Equation 1 for y, because y is an independent variable.

Now you are left with: 4x + 3 (2x +5) = 12

Now, solve the equation: 4x + 15 = 12 6x

Subtract 15 from 12 to get up on the right side. 10x = -3

Now just solve for x

X = -3/10

Now do the same thing for y. find

The replacement is particularly useful when one of the two equations as a variable has been isolated, as equation 2 gives a precise replacementResponse, as opposed to graphical representation. It 'a method to easily perform, but not limited to linear systems, that equation, a variable isolated so if none of the two equations that you are provided with a variable Contain isolated, you can use it in y = x = etc. must change. . Form

Linear combination is a fairly simple way to solve systems of linear equations. This is to eliminate a variable in order to make the system more easily resolved. Linear combination would not work if a usefulThe equation was given which included a variable isolated, could be made, but substitutions would make more sense to use. Here's an example:

First, you must obtain a variable coefficients, which differ only in sign, so that it can be stopped in order to simplify the equations. One way to do this is to multiply both equations or an equation with the correct number.

1 4x + 4y = 6  multiply this equation by 3

2-2x - 3y = -1 to multiply it This equation 4

Result:

1 12x 12 = 18

2-8x - 12y = -4

Now you can add the two equations, and the variable y is eliminated, since their coefficients differ only by sign. Then, to solve the last remaining variable.
1 4x = 14

X 2 = 3.5

Linear combination is useful because it allows for the deletion or removal of one of the variables in the equations for a simple solution. This is simply done by multiplying one or bothEquations with the numbers in the hope that the coefficients of the same number with an opposite sign. Then they added cancel each other out, and can be solved and will.

Another way to solve the equation system has Graphs. Graphically it is great because it gives a visual representation of the system. It is not entirely accurate if done by hand, and can be read. For the graphical solution, two things are necessary: the slope and y-intercept To solve a linear system withGraph, plot the y-intercept of each equation will be made available to graph on a coordinate system. Then use the slope (rise over run) to find the other point. Then draw a line between two points and extend outward up to find the intersection. Then find the ordered pair of this point. Graphical representation is a good way, because it shows visually the equations, but it is wrong much more than other methods.

The last method is to solve the matrices. Matrices area much more simple systems that can solve three or more variables. Usually linear combination could be used, but because it is a monotonous process, matrices may be the best method is. Since the matrices are not divided, must be made by multiplying the inverse matrix will be canceled. The converse (when multiplied) is equal to the identity matrix, which is similar to multiply by "1". The coefficients of a system must be included in an order matrix. Here's an example.

-2x - 3y= -26

3x + 4y = 36

They are matrix A ([A]):

[-2 -3]

[3 4]

They are matrix B ([B]):

[-26]

[36]

The equation should look like this:

[-2 -3] * [X] = [-26]

[3 4] [y] [36]

Then multiply [A] -1 * [B]

[-2 -3] -1 * [-26]

[3 4] [36]

Remember, the matrix multiplication is not commutative, then [B] * [A] -1 an answer other than that [A] -1 *[B]. To solve a problem using arrays of linear equations is time-consuming process. However, it is a good way to complex problems with many variables.

Systems of linear equations can be solved in many ways. Replacement is a nice concise algebraic systems that have an equation with a variable isolated, for example, solve x = 7y - 9 The replacement is very precise, as opposed to graphics, but it is limited to equations that contain a free variable , so that ifTheir equations do not contain them, it is necessary to y = x = o, etc., make the shape of a variable to be isolated. Linear combination is another way of using everything possible to solve systems. Linear combination is a good method because it allows the removal or abandonment of one of the variables in the equations for a simple solution. Graphic representation is preferable to use a method to solve the system because it visually shows the solution. The disadvantage of computer graphics is the inaccuracy of it compared to otherAdd methods. With the matrices is a long boring but very accurate and useful for large systems. Linear equations have many options can be solved with them.