How to Solve Pair of Linear Equations by Substitution Elimination and Graph Method with Examples
A linear equation in two variables is an equation in the form ax+by+c, where a,b, and c are real values and a,b are not equal to zero. We deal with two such equations in a pair of linear equations in two variables. A point on the line representing the equation is the solution of such equations.
An equation is a two-sided expression that has the same sign on both sides. A polynomial is a mathematical expression having non-negative integer powers for the variables. For instance, x4 + 3x3 + 2x9 is a polynomial, whereas x3/5+ 3x0.6 is not. We should be aware of the concept of 'degree' when defining polynomials. The highest power of the variable in the given polynomial is defined as a degree. A linear polynomial is a polynomial with degree one. A polynomial of degree 2 is referred to as a quadratic polynomial, whereas a polynomial of degree 3 is referred to as a cubic polynomial.
How to solve the Pair of Linear equations?
Here, we cannot get a particular solution for this as there is only one condition given, and we have two unknowns. We can rewrite the above equation as:
y = \[\frac {(9-6 x)} {7}\]
The values of y will change in accordance with the values of x. As a result, there is no such thing as a one-of-a-kind answer.
As a result, it is evident that in order to obtain a specific solution of a system of linear equations in two variables, two alternative sets of independent conditions are required.
Representation
Two methods can be used to solve and express the pair of linear equations:
Graphical Approach
The Algebraic Approach
The general formulation of a set of linear equations in two variables, say x and y, is:
a1x + b1y + c1 = 0..............(1)
a2x + b2y + c2 = 0..............(2)
where a1, b1, c1, a2, b2, c2, a12 + b12 = 0 and a22+ b22 = 0 are all real values
When the pair of linear equations is written as a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0, three conditions apply:
1. An inconsistent pair of linear equations is a set of two linear equations in two variables that has no solution.
2. A consistent pair of linear equations in two variables is a pair of linear equations with a solution.
3. A dependent pair of linear equations is a pair of linear equations in two variables that has an infinite solution.
Graphical Representation
If there are two lines in a plane, there are three possibilities:
1. The two lines cross each other,
2. Are parallel to one other
3. Or coincide with each other.
Algebraic Representation
Let's begin by looking at how we might use simultaneous linear equations in our daily lives. In any potential case, a pair of linear equations can be found. Let's pretend you went to the fish market to buy some fish. The fish were available in two sizes. According to the fisherman, the total price of the smaller fish is three times that of the larger fish. In addition, the total amount of money you purchased from your residence is Rs.100. Can you tell me how much you paid for the two different types of fish?
Let's look at this from a mathematical standpoint.
Allow for a pricing difference of Rs.x for the smaller fish and Rs.y for the larger fish.
According to the first condition, x = 3y........... (1)
x + y = 100, according to the second condition ……………(2)
To discover the solution, we must solve both equations and determine the values of x and y. In a graph, the coordinates (x, y) can be simply arranged. However, when the point representing the solution of linear equations has non-integral coordinates, such as (3, 2 7), (–1.75, 3.3), (4/13, 1/18), etc., the graphical method is inconvenient. As a result, we tackle such problems using algebraic approaches.
To solve a pair of linear equations, you can use the algebraic methods listed below:
1. Substitution Techniques
2. Method of Elimination
3. The Method of Cross-Multiplication
A linear equation in two variables refers to the equation in the form of ax + by + c, wherein a,b and c are the real numbers and a,b and c are not equal to zero. However, in the pair of linear equations with two variables, you have to deal with two of these equations. The solution of these equations is a point on the line which represents the equation. An equation is called to be an expression when it has an equality sign on both sides. A polynomial consists of a mathematical expression having powers of the variables as the non-negative integers. When you define polynomials you must know about the concept of the degree. A degree is defined as the highest power of a variable in a given polynomial. A polynomial having a degree 1 is said to be a linear polynomial. A polynomial having a degree 2 is said to be a quadratic polynomial and similarly, a polynomial having a degree 3 is called a cubic polynomial. In this article, we will learn how to solve the pair of linear equations and the pair of linear equations in two variable graphical methods.
Solving the Pair of Linear Equation in Two Variables
You have studied the linear equation in one variable and you would know how to solve it. If there was just one variable and just one equation, you could solve it easily, but in this case, you have two different variables and two different equations. Here you need two different sets of the linear equations for finding out the two different unknowns. If one equation is given to you and you have to solve two variables you will not find a particular solution.
Consider for example the given equations:
4x + 2y = 10 and 5x + 2y = 15
You can solve these simultaneous equations and you can arrive at a particular solution from these equations but on the other hand, if you consider,
7x + 3y = 14
In this case, you cannot get a particular solution as there is only one condition given and you have two different unknowns. You can rewrite the above equation in the following way:
y=14−7x3
Depending on the different values of x, the values of y will also change accordingly. Hence, one unique solution is not possible in this case.
Thus, you can clearly say that to get a specific solution of the systems of linear equations in two variables, you need to have two different sets of independent conditions.
Representation of the Pair of Linear Equation in Two Variables
You can solve the pair of linear equations and represent it in two different ways:
The graphical method
The algebraic method
The general representation of the pair of linear equation in two variables x and y is denoted as:
a1x+b1y+c1=0 and a2x+b2y+c2=0
Here, the numbers a1,b1,c1,a2,b2 and c2 are the real numbers.
Also, a21+b21≠0 and a22+b22≠0
.If the pair of linear equations is given in these two forms of a1x+b1y+c1=0 and a2x+b2y+c2=0, then three different conditions arise:
The pair of linear equations in two variables that have no solution is referred to as an inconsistent pair of the linear equations.
The pair of linear equations in two variables that has a solution is called a consistent pair of the linear equations.
The pair of linear equations in two variables that has an infinite solution is said to be a dependent pair of the linear equations.
Pair of Linear Equations in Two Variables Graphical Method
If two lines are present in a plane, there could be three different possibilities that are as follows:
Two lines intersecting with each other
Two lines parallel to each other
Two lines coinciding each other
You can represent these conditions graphically as follows:
(images will be uploaded soon)
Pair of Linear Equations in Two Variables Algebraic Method
You can solve a pair of linear equations in two variables through the algebraic method in the following ways:
Substitution Method:
You substitute one of the given equations in the other by substituting one of the variables in the form of another. Now, the equation would consist of only one variable and you can then solve it accordingly to get the result.
Elimination Method:
As the name suggests, the elimination method refers to the elimination of one of the variables from the given set of the equations. Solving it would give you the desired result.
Cross-Multiplication Method:
The general form of the pair of linear equations in two variables is given as follows:
a1x+b1y+c1=0 ...(1)
a2x+b2y+c2=0 ...(2)
In this method, you multiply the equation (1) by the coefficient of b2 or a2 and in the equation (2) by that of the b1 or a1 and then eliminate one of the variables and solve. The name is given because the multiplication in the equations having the coefficients of the variables are carried in a cross-multiplication fashion.
Solved Examples
Let us now look at some of the solved examples:
Example 1: Find the values of the two variables that satisfy the following equations:
2x + 5y = 20
3x + 6y = 12
Solution: Using the substitution method to solve the pair of linear equations you have:
2x + 5y = 20 . . . (i)
3x + 6y = 12 . . . (ii)
Multiplying the equation (i) by 3 and the equation (ii) by 2, you get
6x + 15y = 60 . . . (iii)
6x + 12y = 24 . . . (iv)
Subtracting the equation (iv) from (iii),
3y = 36
Hence, y = 12
Substituting the value of the variable y in any of the above-given equation (i) or (ii), you get,
2x + 5(12) = 20
Hence, x = -20
Hence, x = -20 and y = 12 is the point in a plane where the given equations would intersect.
Example 2: Solve the given two equations graphically:
x + y = 16
x - y = 4
Solution: The solutions for each of the equations is as follows:
From the graph, you can find the common point of intersection which is (10, 6).
FAQs on Solve the Pair of Linear Equations in Two Variables
1. What is a pair of linear equations in two variables?
A pair of linear equations in two variables consists of two equations of the form ax + by + c = 0 that are solved simultaneously. Each equation represents a straight line in a plane. The solution is the point where both lines intersect, which satisfies both equations at the same time. For example:
- 2x + y = 5
- x − y = 1
2. How do you solve a pair of linear equations by substitution method?
The substitution method solves a pair of linear equations by expressing one variable in terms of the other and substituting it into the second equation. Steps:
- Solve one equation for one variable.
- Substitute that expression into the other equation.
- Solve the resulting single-variable equation.
- Substitute back to find the second variable.
3. How do you solve a pair of linear equations by elimination method?
The elimination method solves linear equations by eliminating one variable through addition or subtraction. Steps:
- Make the coefficients of one variable equal.
- Add or subtract the equations to eliminate that variable.
- Solve for the remaining variable.
- Substitute back to find the other variable.
4. What is the cross multiplication method for solving linear equations?
The cross multiplication method is a formula-based technique used to solve two linear equations of the form a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0. The formulas are:
- x = (b₁c₂ − b₂c₁) / (a₁b₂ − a₂b₁)
- y = (c₁a₂ − c₂a₁) / (a₁b₂ − a₂b₁)
5. What is the graphical method of solving a pair of linear equations?
The graphical method solves a pair of linear equations by plotting both lines on a coordinate plane and finding their point of intersection. Steps:
- Convert each equation into slope-intercept form if needed.
- Plot at least two points for each line.
- Draw both lines on the same graph.
- The intersection point gives the solution.
6. What are the conditions for a unique solution, no solution, or infinitely many solutions?
The number of solutions of a pair of linear equations depends on the ratios of their coefficients. For equations a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0:
- Unique solution: a₁/a₂ ≠ b₁/b₂
- No solution (parallel lines): a₁/a₂ = b₁/b₂ ≠ c₁/c₂
- Infinitely many solutions (coincident lines): a₁/a₂ = b₁/b₂ = c₁/c₂
7. Can you give an example of solving a pair of linear equations step by step?
Yes, a simple example is solving 3x + 2y = 11 and x + y = 5. Using substitution:
- From x + y = 5, x = 5 − y.
- Substitute into 3x + 2y = 11.
- 3(5 − y) + 2y = 11
- 15 − 3y + 2y = 11
- 15 − y = 11
- y = 4
- Then x = 5 − 4 = 1
8. Why is a pair of linear equations called “linear”?
A pair of linear equations is called linear because the highest power of each variable is 1. There are no terms like x², y², or xy. The general form is ax + by + c = 0, which represents a straight line when graphed. Hence, the word “linear” comes from the fact that the equations form straight lines.
9. What is the general form of a pair of linear equations in two variables?
The general form of a pair of linear equations in two variables is:
- a₁x + b₁y + c₁ = 0
- a₂x + b₂y + c₂ = 0
10. What are common mistakes when solving a pair of linear equations?
Common mistakes in solving a pair of linear equations usually involve calculation and sign errors. Key mistakes to avoid:
- Incorrectly changing signs during elimination.
- Substitution errors while replacing variables.
- Arithmetic mistakes in simplification.
- Not checking if the denominator (a₁b₂ − a₂b₁) equals zero in cross multiplication.
