linear equations in two variables

The linear equations in two variables are a system of equations with a unique solution, no solutions, or infinitely many solutions. A linear system of equations has an ‘n’ number of variables, an important thing to keep in mind while solving linear equations with n number of variables is that there must be n number of equations to solve and determine the value of the variables. The set of solutions obtained on solving the linear equations is a straight line. Linear equations in two variables are algebraic equations that are of the form of y = mx + b, where m is the slope and b is the y-intercept. They are equations of the first order. For example, y = 2x+3 and 2y = 4x + 9 are two-variable linear equations with x and y as variables.

LINEAR EQUATIONS IN TWO VARIABLES-

The linear equations in two variables are of the highest exponents order of 1 and have one, none, or infinitely many solutions for it. The standard form of a two-variable linear equation is ax+ by+ c= 0 where x and y are the variables. The solutions of the equations can also be written in ordered pairs. The graphical representation of linear equations in two variables includes two straight lines which can be either intersecting lines or parallel lines or coincide lines.

FORMS OF LINEAR EQUATIONS IN TWO VARIABLES-

A linear equation in two variables can be in the forms like standard form, intercept form and plane-slope. For example, the equation 2x+3y=9 can also be represented in each of the forms like 2x+3y-9=0 (standard form), y = (-2/3)x + 3 (slope-intercept form), and y – 5/3 = -2/3(x + (-2)) (point-slope form). 

METHODS OF SOLVING LINEAR EQUATIONS IN TWO VARIABLES-

There are various methods to solve a system of linear equations in two variables some are listed below

• The Substitution Method

• The Cross Multiplication Method

• The Elimination Method

The substitution method:

To solve a system of two linear equations having two variables using the substitution method, we have to use the steps given below:

• Step 1: Solve one equation for one variable.

• Step 2: Substitute this in the other equation to get an equation of a single variable.

• Step 3: Solve it for that variable and after you get the value.

• Step 4: Substitute it in any of the equations to get the value of the other variable.

• Example: Solve the following system of equations using substitution method
  x+2y-7=0
  2x-5y+13=0

• Solution: Now Let us solve the equation, x+2y-7=0 for y:
  x+2y-7=0
  ⇒2y=7-x
  ⇒ y=(7-x)/2

• Substitute this in equation, 2x-5y+13=0:
  
  2x-5y+13=0
  ⇒ 2x-5((7-x)/2)+13=0
  ⇒ 2x-(35/2)+(5x/2)+13=0
  ⇒ 2x + (5x/2) = 35/2 – 13
  ⇒ 9x/2 = 9/2
  ⇒ x=1

• Substitute x=1 this in equation y=(7-x)/2:
  y=(7-1)/2 = 3
  Thus, the solution of the given system is x=1 and y=3.

Cross Multiplication Method:

Consider a system of linear equations: a1x + b1y + c1 = 0 and a2x + b2y + c2  = 0

To solve this using the cross multiplication method, we first write the coefficients of x and y.

Here, the arrows indicate that those two coefficients have to be multiplied. Now write the equation by cross multiplying and subtracting the products.

From that equation, we get two equations:

Solving each of these for x and y, the solution of the given system is:

Method of Elimination:

To solve a system of linear equations in two variables using the elimination method, we will use the steps given below to use the method:

• 1: Arrange the equations in standard form which is : ax+by+c=0 or ax+by=c.

• 2: Check if adding or subtracting the equations would result in the cancellation of a variable x and y.

• 3: If not, multiply both equations by either the coefficient of x or y such that their addition or subtraction would result in the cancellation of any one of the variables.

• 4: Solve the resulting single variable equation to get the value of the variable.

• 5: Substitute it in any one of the equations to get the value of the other variable.

Example: Solve the following system of equations using the elimination method.
2x+3y-11=0
3x+2y-9=0

Adding or subtracting these two equations would not result in the cancellation of either x or y. so Let us aim at the cancellation of the variable x. The coefficients of x in both equations are 2 and 3 so Their LCM would be 6. We will make the coefficients of x in both equations 6 and -6 such that the x terms get canceled when we add both the equations.

3 × (2x+3y-11=0)
⇒ 6x+9y-33=0
-2 × (3x+2y-9=0)
⇒ -6x-4y+18=0

Now we will add these two equations:
6x+9y-33=0
-6x-4y+18=0
On adding both the above equations we would get,
⇒ 5y-15=0
⇒ 5y=15
⇒ y=3

Substitute this in one of the given two equations to get the value of x.
2x+3y-11=0
⇒ 2x+3(3)-11=0
⇒ 2x+9-11=0
⇒ 2x=2
⇒ x=1

Thus, the solution of the given system of equations is x=1 and y=3.

PROPERTY OF LINEAR EQUATION-

A linear equation only has either one or two variables. No variable in a linear equation is raised to a power greater than 1 or used as the denominator of any fraction. When you find pairs of values that make a linear equation true and plot those pairs on a coordinate grid, all of those points lie on the same line.

Conclusion:

An equation is said to be a linear equation in two variables if it is written in the form of ax + by + c=0, where a, b & c are real numbers, a and b are not equal to zero since they are the coefficients of x and y.

For example, 10x+4y = 3 and -x+5y = 2 are linear equations in two variables with x and y as variables.

The solution for such an equation is a pair of values, one x, and one y which further makes the two sides of an equation equal.