Algebraic Methods of Solving a Pair of Linear Equations
Various methods of Solving a Pair of Linear Equations are,
- Substitution Method
- Elimination Method
- Cross-Multiplication Method
Substitution Method
In this method, we use one equation to express a variable in terms of the other variable thereby reducing the number of variables in the equation. Then we substitute that expression into the other equation that is given to us.
Example: Solve the following pair of equations with the substitution method.
x + y = 3
3x + y = 16
Solution:
Let’s pick the equation
x + y = 3
x = 3 – y
Substituting the value of x in the other equation,
3x + y = 16
3(3 – y) + y = 16
9 – 3y + y = 16
-2y =7
y = -7/2
Elimination Method
This method is sometimes more convenient than the substitution method. In this method, we eliminate one variable by multiplying and adding equations with suitable constants, this is done to eliminate one variable, and when the equation is left with just one variable, it can easily be solved.
Example: Solve the following equations with the elimination method.
x + y = 3
x – y = 5
Solution:
We have two equations,
x + y = 3 …….(1)
x – y = 5 …… (2)
Adding the equation (1) and (2) to eliminate the variable -y.
2x = 8
x = 4
Substituting the value of x in equation (1)
4 + y = 3
y = -1
Cross-Multiplication Method
This method looks more complex than the other methods, but it is one of the most efficient ways to solve linear equations. Let’s say the two lines whose equation is:
a1x + b1y + c1 = 0
a2x + b2y + c2 = 0
In this cross-multiplication method,
Cross-Multiplication Method
The solution is given by:
x/(b1c2-b2c1) = y/(c1a2-c2a1) = 1/(a1b2-a2b1)
Example: Solve the following equations with the cross-multiplication method.
2x + 3y = 46
3x + 5y = 74
Solution:
a1 = 2, a2 = 3, b1 = 3, b2 = 5, c1 = 46 and c2 = 74
x/(b1c2-b2c1) = y/(c1a2-c2a1) = 1/(a1b2-a2b1)
x/(3-(-74)-5(-46)) = y/(-46(3)-(-74)2) = 1/(2(5)-3(3))
x/8 = y/10 = 1/1
x = 8 and y = 10
Pair of Linear Equations in Two Variables
Linear Equation in two variables are equations with only two variables and the exponent of the variable is 1. This system of equations can have a unique solution, no solution, or an infinite solution according to the given initial condition. Linear equations are used to describe a relationship between two variables. Sometimes in some situations, we don’t know the values of the variables we want to observe. So, then we formulate the equations describing how they behave and solve them. The Number of equations obtained should be equal to the number of variables.
Let’s learn about the Pair Of Linear Equation In Two Variables and their solution in this article.
Table of Content
- Pair of Linear Equations In Two Variables
- What are Pair of Linear Equations in Two Variables?
- Pair of Linear Equations in Two Variables Formulas
- Representation of Pair of Linear Equation in Two Variables
- Graphical Representation
- Algebraic Methods of Solving a Pair of Linear Equations
- Pair of Linear Equations in Two Variables Class 10 Extra Questions
- Pair of Linear Equations in Two Variables Solutions