Systems of Linear Equations in Two Variables

Systems of Linear Equations in Two Variables:


The basic idea behind the addition/elimination technique is to multiple one or more equations by a constant therefore whenever they are added altogether, one of the variables removes. Then you have one equation with one variable and you can resolve for that variable.

a) Select a variable to remove. Generally the variable which can be removed by multiplying by smaller numbers is the best choice.

b) Multiply one or both equations by the constant and hence the least common multiple of the coefficients on variable to be removed is obtained. Care must be taken and hence one coefficient becomes negative and the other positive.

c) Add up the two equations together thus the variable is eliminated.

d) Solve the resultant equation for the remaining variable.

e) Back-substitute that value to the one of two original equations to determine the remaining variable.

f) Check your answer to the other equation.

Since an alternative to step e, and this is very helpful whenever the answer is a fraction or decimal value and not pleasing to work with, you can go via the elimination procedure again with other variable, and then you do not have to work with fractions till the check procedure.

Graphical Interpretation of Solutions:

You might have no solution, one unique solution or many solutions whenever solving a 2x2 system of the linear equations.

Exactly One Solution:

a) Intersecting lines.
b) Consistent System (consistent signifies that there is a solution, there is no contradiction).
c) Independent System (that is, the value for y does not depend on what x is).

Write your outcome or answer as the set having the ordered pair {(x, y)} (replace x and y with real values).

This would be equivalent to a conditional system, some values of x and y makes it true, whereas others do not.

Let consider the system of linear equations 3x + 2y = 17 and 2x - y = 2. The solution is x = 3 and y = 4, therefore you write the answer as {(3, 4)}.

No Solution:

a) Parallel lines.
b) Inconsistent System (contradiction).
c) Independent or Dependent does not apply as there is no solution.

Write down your answer as ‘no solution’, that is, the symbol for empty or null set, Ø, or the empty set { }, however do not write it as the set having null set {Ø}. Be cautious if you're someone who slashes their zeros.

This case takes place if both the variables eliminate, and you are left with a false statement.

Let consider the system of linear equations 3x - 2y = 3 and -3x + 2y = 2. Whenever you add them altogether, you get 0 = 5 that is a contradiction. Thus, the answer is ‘no solution’, { }, or Ø

Many Solutions:

a) Coincident lines (similar line).
b) Consistent System (identity).
c) Dependent System (that is, the value for y will mainly depend on what x is, it is not always the similar value).

Write down your answer as one of the two equations. Don’t say ‘many solutions’ or ‘all real numbers’. All real numbers will not work. At First, the solutions are of ordered pairs, not individual x or y coordinates. Secondly, not each and every point works, only such on the line work.

You might as well write your answer in the parametric form. This will be the favored method for higher ordered systems; therefore you might as well learn it now.

This case takes place if both the variables eliminate, and you are left with the true statement.

Let consider the system of linear equations 3x - 2y = 3 and -3x + 2y = -3. Whenever you add them altogether, you get 0 = 0 that is always true. Thus, any values of x and y which satisfy equation are the solutions. You would write your solution as:

3x - 2y = 3
{(x, y)|3x - 2y = 3}
x = t, y = 3/2 t - 3/2 (this is termed as parametric form)

Making ‘nice’ problems:

Ever think why most of all the problems from algebra textbook come out with nice integer answers? Is it as life is that way? Certainly not, it is since the problems are contrived to encompass nice answers.

How do you obtain a made up problem to encompass nice answers? The answer is that you do not. You begin with the answer and work back-wards.

Let us state that we wish the answer to be (3, -2).

Make up something for the left side of equation, say 2x-5y. Then, plug in x = 3 and y = -2, therefore 2(3) - 5(-2) = 6 + 10 = 16. Then first equation is 2x-5y = 16.

Now, repeat the procedure again with a distinct left side, state 3x + 2y. Well, 3(3) + 2(-2) = 9 - 4 = 5, therefore 3x+2y=5.
Your system of linear equations is 2x - 5y = 16 and 3x + 2y = 5.

You can make up something for left hand side; just make sure that the right hand side is what you obtain whenever you plug such values in. You will as well require having two equations when there are two variables, one equation when there is only one variable, three equations when there are three variables and so on.

Least Squares Regression Model:

Till this point, whenever we have found the linear regression model, we have just utilized the functions on calculator to get the outcomes or results, and it has been fairly simple and painless. Now we learn that calculator is in fact solving a system of linear equations to get the model.

Summation notation:

The capital Greek letter sigma signifies for sum. Generally, there is an index with a beginning point (k = 1) written beneath the sigma and an ending point (n, meaning k = n) written above the sigma. Then, each and every variable will contain a subscript to let you know that it is the function or sequence which depends on the value of index, k.

Now, we use a short hand notation to keep the things simpler and easier to keep in mind. Rather than writing it the way the book does, just use a sigma symbol and then what we wish to total.

2013_Least regression.jpg

Keep in mind that sigma signifies sum, therefore sigma x signifies add up all the x's.

In statistics, we like to simplify the things, and get a little sloppy, and drop all the index stuff and merely know that it applies to all points. In the notation above, we will use the form on the right. ∑x just signifies add up all the x values. Not too bad whenever you look at it that manner.

Linear Regression:

Let consider the linear model y = ax + b. The values for a and b can be determine by solving this system of the linear equations.

b∑1 + a∑x = ∑y
b∑x + a∑x2 = ∑xy

Note that each term in the second equation consists of one more x in it than the corresponding term in first equation. This pattern will recur whenever we do quadratic regression.

We will add up each and every variable in the summation for each distinct point. The first summation is the sum of 1. Therefore, if you add up 1 for each and every point, you will simply contain the number of points. Other values are the sum of x's, the sum of the y's, the sum of the squares of x's, and the sum of products of x's and y's.

Writing such ordered pairs in a columnar table, and then adding up columns for the x2 and xy will help. After you resolve the system of linear equations, replace the values for a and b to the equation y = ax + b to obtain the model.

Find out the equation of linear model which best fits the points (2, 3), (5, 2), (6, 1), and (8,-1).

Set up a table with columns for x, y, x2, and xy.

376_linear regression table.jpg

The numbers in the bottom row symbolizes the sums which go to the system. As there are 4 points, the ∑1 = 4.

The system of linear equations to resolve is 4b + 21a = 5 and 21b + 129a = 14. Whenever you resolve that, you obtain a = -49/75 and b = 117/25.

Whenever you stick such back to the model, you obtain y = -49/75 x + 117/25.

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