Linear Equations in Two Variables

Linear Equations in Two Variables. In previous tutorial editions in mathematics, we discussed what is meant by a linear equations in one variables which is then forwarded by an example problem and its completion steps.

Therefore in this tutorial, we will continue the discussion of linear equations, but by using two variables commonly called the two variable linear equations. As usual, we will start with a basic understanding of what is meant by the linear equation in two variables definition, then where the difference with the linear equations of one variable then we enter the new exercise problem and its completion steps.

What is a Linear Equation in Two Variables definition ?

The linear equation in two variables is a linear equation which has two variables, with the rank of each variable being one. Linear Equations Two Variables have a common form :

 ax + by = c where x and y is variable and a, b, c ∈ R (a ≠ 0, b ≠ 0).

Example of a Linear Equation in Two Variables
Which of the following is considered a linear inequality of one variable :
a.  3x - 4y  = 12
b. 7w - 4z + 5 = 3w + 2z + 10
c. 2x - 2 = 10
d. 𝑝2 − 2𝑝 + 1 ≤ 0

Discussion:
a. The variable in the equation: 3x - 4y  = 12 is the x and y and the power of each variable is 1. Then the equation is included in the linear equations in two variables.

b. The variables in the equation : 7w - 4z + 5 = 3w + 2z + 10 are w and z and each variable is one rank. Since it has two variables (w and z) and one rank, it includes two linear equations.

c. The variables in the equation: 2x - 2 = 10 are x and one rank. Since there is only one variable, then this equation does not include a two-linear equation .

d. The variable in the equation : 𝑝2 − 2𝑝 + 1 ≤ 0 is p one and two. Since there are only two variables that are only distinguished from the rank, it is considered a variable of one type or one only. Thus it is considered not a linear equation in two variables .

See Also : Transpose Of a Matrix Problems and Solving

linear equations in two variables worksheets

Here are some ways that you can solve a two-linear equation:

1. Substitution method

Substitution method is done by replacing a variable with variable from other equation.

Example :
2x - y = 6 ..................... (i)
x + y = 3 ......................(ii)

First Step
Changed one of the equations in the form x = .... Or y = ....
From equation (i), we can obtain :

2x -6 = y
       y = 2x -6

The second step
Substitute the equation above to equation (ii) so that the value of x :

x + (2x – 6) = 3
     3x – 6  = 3
          3x = 3 + 6
          3x = 9
           x = 3

The Step Three
value of x = 3 is fed to equation (i) or to equation (ii).
Let x = 3 be substituted into equation (ii), so that the value y is obtained :

x + y = 3
3 + y = 3
    y = 3 - 3
    y = 0

If its x = 3 is inserted into equation (i), so the value of y :

2x  – y = 6
2.3 - y = 6
  6 – y = 6
      y = 6-6
      y = 0

Thus there is no problem whether put into equation (i) or (ii) then the value of y obtained remains the same.
so, the solution is x = 3 and y = 0, written HP = {(3.0)}

2. Method of elimination

The elimination method is done by eliminating one of the variables.

Example :

2x – y = 6 ……(i)
x  + y = 3 ……(ii)

Initial Step
We can eliminate any of the variables, either the x or y variables. To eliminate the variables, consider the two equations, how many times so that if added or subtracted then there are missing variables.

In this step, we want to remove the x variable first. We know the value of 2x in equation (i) and the value of x in equation (ii). In order to be lost then we multiply one (x 1) in equation (i) and times two in equation two (ii), then the product is subtracted :

2x – y = 6 |x 1| ⇔ 2x – y  = 6
 x + y = 3 |x 2| ⇔ 2x + 2y = 6

2x -  y = 6
2x + 2y = 6
___________ _
     -3y = 0
       y = 0

Step Two
We will eliminate the variable y
If we look at the equation (i) has the value -y and equation (ii) has the value y, then the two equations can be directly added to missing the y variable.

2 x – y = 6
  x + y = 3
___________ +
     3x = 9
      x = 3

so, the solution is x = 3 and y = 0, written HP = {(3.0)}

3. Mixed Method (Merging of Substitution Method + Elimination Method)

This method is accomplished by combining the method of elimination and substitution methods.

Example :

2x – y = 6 ……(i)
x  + y = 3 ……(ii)

The First Step
We do the elimination method by removing the variable x
2x – y = 6 |x 1| ⇔ 2x – y = 6

 x + y = 3 |x 2| ⇔ 2x + 2y = 6

2x -  y = 6
2x + 2y = 6
___________ _
     -3y = 0
       y = 0

Step two
In this second step, the substitution method is done by entering the value into an equation.

Enter the y value that can be given to equation (i) or to equation to (ii). Suppose we enter into the equation (i), then :

2x – y = 6
2x - 0 = 6
    2x = 6
     x = 3

so, the solution is x = 3 and y = 0, written HP = {(3.0)}

4. Graph Method

In the graph method, we have to illustrate the graph of the two equations. The intersection between the two graphs is taken as the solution.

For example, we have two equations:

x + y  = 4   ......(i)
x + 2y = 6   ......(ii)

First step
We will draw a graph for equation (i): x + y = 4.
To illustrate the graph, of course we must look for the intersection points in x and in y, so that :

If x = 0,
then: x + y = 4 0 + y = 4 y = 4
=> the cutoff at y (0, 4) If y = 0,
then: x + y = 4 x + 0 = 4 x = 4,
=> the intersection point in x (4, 0)
So the intersection point of the equation x + y = 4 is (0.4) and (4.0)


Second step
We will draw the graph for equation (ii): x + 2y = 6

If x = 0, then:
 x + 2y = 4 0 + 2y = 4 y = 2
 => the cutoff at y (0, 2)
if y = 0, then: x + 2y = 6 x + 0 = 6 x = 6,
=> x (6, 0)
So the cutoff point of the equation x + 2y = 6 is (0.2) and (6.0)


Third Step
From the intersection point (i) and equation (ii) we draw the graph as shown below:

From the picture above, the coordinates of the intersection of the two lines are (3, 1). Thus, the set of completions is {(3, 1)}.

Exercises

Problem No.1

Dina Age 7 years younger than age Desi. Their age is 43 years. How old is Dina and Desi?

Solution:

For Example:

Age Dina = x Age Desi = y
Then: Age Dina 7 years younger than Desi's age can be made into an equation:
y - x = 7 ... (1)
Their 43 years can be made into a equation:
 x + y = 43 ... (2)
Equation (1):
y - x = 7 y = 7 + x
Then substitute y = 7 + x into equation
 (2) x + y = 43 x + 7 + x = 43 2x + 7 = 43 2x = 43 - 7 2x = 36 x = 18
Thus, Dina's age is 18 years and the age of Dec 25 years.

Problem No.2

Be discovered:


4x + 3y = 34 ... Equation (i)
5x + y = 37 ..... Equation (ii)

Find the values ​​of x and y by the method of elemination ?  

Solution:
First Step
To eliminate the variable y, then equation number 1 multiplied by 1 and equation number 2 multiplied by 3. Both equations are subtracted for y missing variables.

4x + 3y = 34 |x 1| ⇔  4x + 3y = 34 
5x +  y = 37 |x 3| ⇔ 15x + 3y = 111

 4x + 3y = 34
15x + 3y = 111
_____________ - 
    -11x = -77
       x = 7

Step Two
To find the value of y, equation number 1 is multiplied by 5 and equation number 2 is multiplied by 4. Both equations are reduced in order for the variable x to be lost.

4x + 3y = 34 |x 5| ⇔  20x + 15y = 170
5x +  y = 37 |x 4| ⇔  20x +  4y = 148

20x + 15y = 170
20x +  4y = 148
_____________ - 
      11y = 22
        y = 2

So Value x = 7 and value y = 2.

So first our discussion of Linear Equations in Two Variables may be useful