# Linear Equations in Two Variables

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## Pairs of Linear Equations in Two Variables

A system of a linear equation consists of two or more equations which share a common solution. In a system of linear equations, each equation can be represented by a straight line and the solution is the point where the two or more lines intersect. The concept of the linear equation in mathematics has many implications in real life. Here, we have attempted to give you an idea of the linear equation and how to solve it through solved examples and explanation.

### Linear Equation in Two Variables Definition

If you are asked to define linear equation in two variables, the answer will be - a pair of linear equations in two variables is a set in which all the equations are considered concurrent. To solve linear equations in two variables questions; for a unique solution, one has to find the values of the variable that is consistently true for all the equations. In some cases, linear systems may not have a solution while in other cases, some may have an infinite number of solutions. For finding a unique solution, the number of unknown variables should be equal to the number of equations. Linear equations in two variables have direct application in determining the geometry of lines. As a linear equation represents a straight line, plotting graphs and observing the point of intersection helps to solve for the variables.

### Linear Equation in Two Variable Formula

If a, b, and r are real numbers and both are not equal to 0, then ax + by = r illustrates a linear equation in two variables. The two variables of the equation are denoted by x and y. The numbers a and b represent the coefficients.

• The standard form of linear equation in two variable: ax+by = r.

• The number ‘r’ is called the constant in the above equation.

• A linear equation in two variables has three entities as denoted in the following example:

• 10x - 3y = 5 and 2x + 4y = 7 are representative forms of  linear equations in two variables.

### Solving Linear Equations in Two Variables

While solving a linear equation in two variables, one must always abide by the following rules.

The solution of a linear equation is not altered when:

(i) the same number is added to (or subtracted from) L.H.S. and R.H.S. of the equation.

(ii) multiplication or dividing both L.H.S. and R.H.S. by the same non-zero number.

Every linear equation in one variable has a unique solution. But a pair of linear equations have two solutions i.e. one for x and the other for y, satisfying both the equations.

• ### Unique Solution

If a given set of linear equation intersects at a point, the solution will be unique for both the equations. In order to get a unique solution for a set of equations, the slopes of the lines should differ from each other. Suppose m1 and m2 are the two slopes of two lines having equation in two variables. In that case, when m1 ≠ m2, we can expect a unique solution.

• ### No Solution

If a pair of two variable equations have equal slopes i.e. m1 = m2, the lines will be parallel to each other. Thus, due to the lack of intersection, they will have no solution.

• Unique and Infinite Solutions

If a pair of linear equation is consistent then the lines will have unique solutions as well asinfinite solutions i.e. they will be intersecting or coincident

For Example: Find the value of variables which satisfies the following equations:

8x + 7y = 38 and 3x - 5y = -1.

Solution: Using the method of substitution to solve the pair of linear equation, we have:

8x + 7y = 38…………………….(i)

3x - 5y = -1……………………..(ii)

Multiplying equation (i) by 5 and (ii) by 7, we have:

40x + 35y = 190…………………….(iii)

21x - 35y = -7 ……………………..(iv)

Adding equation (iv) and (iii), we have:

61x  = 183

⇒ x = 3

Substituting the value of x in either equation (i) or (ii), we have

3 x 3 – 5y = -1

⇒ y = 2

Therefore, x = 3 and y = 2 is the point where the given lines intersect.

### Plotting Graph of Linear Equations in Two Variables

After knowing how to solve a pair of linear equations algebraically, let us represent an equation on a coordinate plane. These equations tend to have infinitely many solutions. The solutions of the linear equation: 3x + 2y = 6 can be illustrated in the form of a table as follows by writing the values of y below the corresponding values of x :

 x 0 1 2 y 3 $\frac{1}{2}$ 0

On plotting the points (0, 3), (1, $\frac{1}{2}$) and (2, 0) a graph paper and joining them we will see that it forms a straight line. This is called the graph of the linear equation. The graph is obtained by joining two points corresponding to two solutions. Every point on the line satisfies the equation of the line and acts as the solution to the equation.

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### Did You Know

In our everyday life, we come across several situations where we can use linear equation method to solve a problem. In fact, any situation where one quantity is unknown can be represented by a linear equation! You want to figure out income over time or predict the profit of your startup or calculate mileage - you have to use a linear equation in two variables to find out the unknown quantity.

FAQ (Frequently Asked Questions)

Q1. Mrs Ahuja Lost Her Wallet that had 50 Rupees and 100 Rupee Notes that Amount to 1500 in a Mall. Represent the Composition of the Wallet as an Equation and Draw the Graph.

Ans: Let the number of 50 rupee notes be x

And the number of 100 rupee notes be y

∴ We have 50x + 100y= 1500

⇒ x + 2y = 30

(dividing each term by 50)

2y = 30 – x

⇒ x = 30 – 2y

When y = 10

x = 30 – 20 = 10

When y = 8

x = 30 – 16 = 14

When y = 9

x = 30 – 18 = 12

 x 14 12 10 y 8 9 10

Now, let us plot the points (14,8), (12,9) and (10,10) on the graph to see whether it forms a straight line.

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Thus the straight line indicates the stability of the solution.

Q2. Draw the Graph of the Equation 2x + 3y = 12 and Find the Coordinates of the Point:

(i) Where Y-coordinate is 3

(ii) Where X-coordinate is −3

Ans: We are given,

2x + 3y = 12

We get,

⇒ 2x = 12 – 3y

(dividing each term by 2)

⇒ x =  6 – 3/2 y

When y = 0,

x = 6

When y = 4,

x = 0

 x 0 6 y 4 0

On plotting the above points B (0, 4) and C (6, 0) we get the graph for the equation.

(i) A(3/2 , 3) = point with y coordinate 3

(ii)  D (-3, 6) = point with x coordinate -3

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