WORKSHEET ON Simultaneous linear equations:
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How to solve for variables x and y in a system of simultaneous linear equations:
In this worksheet, we will learn the two methods of solving for variables such as x, y, z and others from a system of simultaneous linear equations.
Let us proceed:
SOLVED PROBLEM 1:
First, let us write down a system of linear equations in two variables, say x and y as follows:
2x + 4y =10 and
3x + 2y = 3
Now, there are two different methods for solving for two variables x and y from a system of linear equations
They are:
The Substitution method and
the Elimination method.
In this worksheet, we will discuss both the methods of solving for two variables x and y from a system of two or more linear equations.
First let us the discuss the elimination method for solving variables from a system of linear equations:
To find the values of the variables x and y from the system of the linear equations, let us first write down the above equations once again below:
2x + 4y =10 and
3x + 2y = 3
To find the value of the variable x, if that is what you want to find first, then multiply the first linear equation with the coefficient of the variable y in the second equation and multiply the second linear equation with the coefficient of the variable y in the first equation.
This is the process to eliminate the variable y from the two linear equations facilitating in solving for x:
And in case, if you wish to find the value of the variable y first, then apply the procedure described above in italics to eliminate x which will then give us a system of equations in only the variable y to find the value of the variable y.
Come on, let’s get on with the implementation of the above elimination method: :
Write down the system of linear equations below:
2x + 4y = 10
3x + 2y = 3
Let us apply the above elimination method to find the value of the variable ‘x’ first:
To do that:
Multiply the first equation with the coefficient of the variable ‘y’ in the second linear equation which is 2 and similarly multiply the second equation with the coefficient of the variable y in the first equation which is 4.
The above elimination method of multiplying the two linear equations with the respective factors: 2 and 4 is displayed:
(2x + 4y =10) × 2
(3x + 2y = 3) × 4
Now, rewrite the original equations with the respective factors multiplied as shown below:
4x + 8y = 20
12x + 8y = 12
Now, subtract the second equation from the first equation (the new ones obtained after multiplying the equations with the respective factors).
To subtract the second equation from the first equation is same as adding the terms of the first equation to the terms of the second equation with the signs of the second equations terms reversed, as below:
(4x + 8y - 20) – (12x + 8y - 12) = 0
4x – 12x + 8y -8y -20 + 12 = 0
-8x – 8 = 0.
In the equation -8x – 8 = 0, there is no term having the variable y. that means all terms in the two linear equations having the variable y have disappeared; because we have eliminated the variable y by applying the elimination method.
-8x = 8, i.e. x = -1.
Thus the value of the variable x satisfying the system of the above two linear equations is -1.
We can now, apply the same method of eliminating variable x to pave way for finding the value of the other variable ‘y’.
Note:
There is no need to eliminate the variable x from the system of the linear equation in order to find the value of the variable y from the system of the above linear equations as we can substitute the value of the variable x = -1 in any one of the above two equations. But to just learn the method of eliminating one variable to find the value of the other variable through the elimination method, we are going through the following discussion of eliminating the variable x for finding the value of the variable y
First, let us write once again the original system of the linear equations (before multiplying the linear equations with necessary factors for eliminating the variable x to find the value of the variable y):
2x + 4y = 10
3x + 2y = 3
Now let us set the process of elimination method into action by trying to eliminate the variable x for finding the value of the variable y by multiplying the first of the above system of linear equations with the coefficient of x in the second equation and the second with the coefficient of x in the first equation as below:
(2x + 4y = 10) × 3 which becomes 6x + 12y = 30 and
(3x + 2y = 3) × 2 which becomes 6x + 4y = 6
Let us write below the above two linear equations with the necessary factors multiplied:
6x + 12y = 30
6x + 4y = 6
Now proceed to eliminate the variable x from the system of the above linear equations in order to pave way for finding the value of the variable y (as part of the standard process of elimination method) by subtracting the terms of the second equation from those of the first equation (this can be accomplished by adding the terms of the first equation above: 6x + 12y = 20 to the terms of the second linear equation with their signs reversed from positive to negative and vice-versa as follows:
(6x + 12y – 30) – (6x + 4y – 6 ) which then becomes simplified to
6x – 6x + 12y – 4y – 30 + 6 = 0
i.e. 8y – 24 = 0
Note that in this equation 8y – 24 = 0, there is no term containing the variable x because we have eliminated x for finding y.
i.e. 8y = 24. i.e. y = 24/8 = 3
thus the values of the values of the two variable x and y obtained by the process of elimination method are -1 and 3 respectively.
