Cross Multiplication Pair of Linear Equations in Two Variables

Introduction

Cross multiplication method is used to solve linear equation in two variables. An algebraic equation in the form of $\mathrm{y\:=\:mx\:+\:b}$ is called a Linear equation. In this equation m is the slope, b is the intercept. x and y is the distance of this line from x-axis and y-axis. The solution of each equation represents a point on the line.

An equation in the form of $\mathrm{ax\:+\:by\:+\:c\:=\:0}$ where a, b, c are real numbers, and the values of both a and b are not zero. Now this equation will be called the linear equation in two variables x and y. The solution of these kinds of equations gives a pair of solutions for x and y.

Pair of Linear Equation in two variables

Two linear equations are in the same two variables x and y, these kinds of equations are called a “pair of linear equations in two variables”.

Algebraically the general equation for the pair of linear equation in two variables x and y is- $\mathrm{a_{1}x\:+\:b_{1}y\:+\:c_{1}\:=\:1}$ and, $\mathrm{a_{2}x\:+\:b_{2}y\:+\:c_{2}\:=\:0}$ where 𝑎1, 𝑏1, 𝑐1,𝑎2, 𝑏2, 𝑐2 are all real numbers and the value of $\mathrm{{a_{1}^{2}\:+\:{b_{1}}^{2}\:\neq\:0\:,\:{a_{2}^{2}\:+\:{b_{2}}^{2}\:\neq\:0}}}$.

The Geometrical representation of a linear equation in two variables is a straight line.

A pair of linear equation in two variables represents two straight lines and hold the following three possibilities −

• The two lines will intersect at one point-

• The two lines will not intersect which means they are parallel-

• The two lines will coincident-

Consistency and Inconsistency

Let, $\mathrm{a_{1}x\:+\:b_{1}y\:+\:c_{1}\:=\:0}$

$\mathrm{a_{2}x\:+\:b_{2}y\:+\:c_{2}\:=\:0}$ be two systems of linear equation

A pair of linear equations that has no solution is called an inconsistent pair of linear equations. A pair of linear equations in two variables that have a solution is called a consistent pair of linear equations. A corresponding pair of linear equations has infinite and distinct common solutions, and such a pair is called a dependent pair of linear equations in two variables that is always consistent.

If two straight lines show the following condition −

• Two lines intersecting at a single point give the unique solution-

$\mathrm{\frac{a_{1}}{a_{2}}\neq\:\frac{b_{1}}{b_{2}}}$ This shows that the pair is consistent

• Two lines may be parallel to each other and gives no solution-

$\mathrm{\frac{a_{1}}{a_{2}}\:=\:\frac{b_{1}}{b_{2}}\:\neq\:\frac{c_{1}}{c_{2}}}$ This shows that the pair is inconsistent.

• Two lines may be coincident giving infinite solutions-

$\mathrm{\frac{a_{1}}{a_{2}}\:=\:\frac{b_{1}}{b_{2}}\:=\:\frac{c_{1}}{c_{2}}}$ This shows the pair is dependent (consistent).

So, it is clear that number of solutions of a pair of linear equation ranges from 0 to ∞ .

Solving a pair of Linear Equation: Cross Multiplication method

It is an algebraic method to solve a pair of linear equations in two variables

Let’s discuss how this method works for any pair of linear equations in two variables. The linear equation of two variables are

$$\mathrm{a_{1}x\:+\:b_{1}y\:+\:c_{1}\:=\:0\:\:\:\rightarrow\:(1)}$$

$$\mathrm{a_{2}x\:+\:b_{2}y\:+\:c_{2}\:=\:0\:\:\:\rightarrow\:(2)}$$

To determine the value of x and y from the above two equations we have to follow these steps −

Step 1 − Multiply equation (1) by $\mathrm{b_{2}}$ and equation (2) by $\mathrm{b_{1}}$, we will get

$$\mathrm{b_{2}a_{1}x\:+\:b_{2}b_{1}y\:+\:b_{2}c_{1}\:=\:0\:\:\:\rightarrow\:(3)}$$

$$\mathrm{b_{1}a_{2}x\:+\:b_{1}b_{2}y\:+\:b_{1}c_{2}\:=\:0\:\:\:\rightarrow\:(4)}$$

Step 2 − By subtracting equation (4) from (3), we get −

$$\mathrm{(b_{2}a_{1}\:-\:b_{1}a_{2})x\:+\:(b_{2}b_{1}\:-\:b_{1}b_{2})y\:+\:(b_{2}c_{1}\:-\:b_{2}C_{2})\:=\:0}$$

$$\mathrm{i.e..,(b_{2}a_{1}\:-\:b_{1}a_{2})x\:=\:(b_{1}c_{2}\:-\:b_{2}C_{1})}$$

so,$\mathrm{x\:=\:\frac{b_{1}c_{2}\:-\:b_{2}c_{1}}{a_{1}b_{2}\:-\:a_{2}b_{1}}}$ and the value of $\mathrm{b_{2}a_{1}\:-\:b_{1}a_{2}\:\neq\:0\:\:\:\rightarrow\:(5)}$

Step 3 − Substitute the value of x from the equation (5) in (1) or (2), to get

$$\mathrm{y\:=\:\frac{c_{1}a_{2}\:-\:c_{2}a_{1}}{a_{1}b_{2}\:-\:a_{2}b_{1}}\:\:\:\rightarrow\:(6)}$$

We can write the solution given by equation (5) and (6) in the following way-

$$\mathrm{\frac{x}{b_{1}c_{2}\:-\:b_{2}c_{1}}\:=\:\frac{y}{c_{1}a_{2}\:-\:c_{2}a_{1}}\:=\:\frac{1}{a_{1}b_{2}\:-\:a_{2}b_{1}}\:\:\:\rightarrow\:(7)}$$

Equation (7) can be written as the following diagram which clearly shows why this method is called a “cross multiplication method”.

Other Methods:

• Substitution − We Substitution the value of one variable by expressing it in terms of another variable to solve the pair of given linear equations . This method is known as the substitution method.

• Elimination − We eliminate one variable first to get the linear equation in one variable

• Graphical − The graph of a pair of linear equations in two variables is represented by two lines. When two lines intersect it gives a unique solution, when they coincide it gives infinite solutions, and if they are parallel it gives no solution.

• Matrix method − Write down the pair of linear equation in a matrix and find the three Determinant D, Dx, Dy for 2× 2 system. These three determinants are coefficient, x, and y determinants. Now we have our determinants we must find the value of x and y by this formula $\mathrm{x\:=\:\frac{D_{x}}{D}\:,\:\frac{D_{y}}{D}}$

Solved Example

1) Find the value of x and y from the given equation

$$\mathrm{4x\:+\:3y\:-\:6\:=\:0}$$

$$\mathrm{and\:x\:+\:3y\:-\:3}$$

Answer −

$$\mathrm{\frac{x}{-9\:-\:(-18)}\:=\:\frac{y}{-6\:-\:(-12)}\:=\:\frac{1}{12\:-\:3}}$$

$$\mathrm{\frac{x}{9}\:=\:\frac{y}{6}\:=\:\frac{1}{6}}$$

$$\mathrm{Taking\:,\:\frac{x}{9}=\:\frac{1}{6}\:and\:\frac{y}{6}\:=\:\frac{1}{6}}$$

$$\mathrm{x\:=\:\frac{3}{2}\:and\:y\:=\:1}$$

2) Three bikes and two cycles cost Rs.1650. Five bikes and three cycles cost 2650. Find the cost of two cycles and two bikes?

Answer − Let the cost of one bike = Rs x

and the cost of one cycle = Rs y

according to the question $\mathrm{3x\:+\:2y\:=\:1650\:\:\:-(1)}$

$\mathrm{\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:5x\:+\:3y\:=\:2650\:\:\:-(2)}$

solving (1) and (2) by cross multiplication method, we get

$$\mathrm{\frac{x}{2\times\:2650\:-\:3\times\:1650}\:=\:\frac{y}{1650\times\:5\:-\:2650\times\:3}\:=\:\frac{1}{3\times\:3\:-\:5\times\:2}}$$

$$\mathrm{\frac{x}{5300\:-\:4950}\:=\:\frac{y}{8250\:-\:7950}\:=\:\frac{1}{9\:-\:10}}$$

$$\mathrm{\frac{x}{350}\:=\:\frac{y}{300}\:=\:1}$$

$$\mathrm{x\:=\:350\:and\:y\:=\:300}$$

cost of 1 bike = Rs.350

cost of i cycle = Rs 300

Cost of 2 bikes and 2 cycles

$$\mathrm{=\:2\times\:350\:+\:2\times\:300}$$

$$\mathrm{=\:700\:+\:600}$$

$$\mathrm{=\:Rs.1300}$$

Conclusion

In this tutorial we have discussed the pair of linear equations in two variables. And learnt various methods to solve the pair of linear equations. In which Cross multiplication method is the easiest to use and solve.

FAQs

1. What is a linear equation?

Equation that is written in the form of $\mathrm{y\:=\:mx\:+\:b}$.

2. What do you understand by the term pair of linear equations in two variables?

Two linear equations having the same variable x and y in it are called a pair of linear equations in two variables.

3. Which pair of linear equations will be called Inconsistent?

Pair of linear equations having no solutions is called Inconsistent.

4. What methods do we use to solve the pair of linear equations in two variables?

Cross multiplication, substitution, matrix, elimination and graphical methods.