In view of the coronavirus pandemic, we are making LIVE CLASSES and VIDEO CLASSES completely FREE to prevent interruption in studies
Business Mathematics and Statistics > Business Mathematics > Simultaneous Linear Equations up to Three Variables
Business Mathematics

Simultaneous Linear Equations up to Three Variables

The System of Equations: Suppose you went to the market and bought trousers and a pair of shirts. Your friend asks you the cost of each of the item. You tell him that the cost of the trousers is three hundred more than the cost of one shirt. The total amount you paid to the shopkeeper was one thousand and fifty rupees.

Obviously, your friend can calculate the prices of each of the item. How does he do so? He simply translated the statements into a system of equations and got the answer. But what will happen if we have more items and the number of the simultaneous equations is also more than two? In this section, we are going to learn about the system of equations and the methods to solve them.

Suggested Videos

Introduction to Logarithms
Introduction to Quadratic Equations
Types of Matrices
Addition and Subtraction of Matrices


The System of Simultaneous Linear Equations

An equation is a mathematical statement showing the relationship of equality. It is a general form of showing the relationship by using a combination of letters, numbers, and symbols. The constant values have the fixed values like 12, 5, −4 etc. and the variables denote the unknowns. They are represented by English letters like a, b, c, x, y, z etc.

Browse more Topics under Business Mathematics

I. System of Equations in Two Variables

A relationship between two unknown values is shown by an equation. A set of values for the two unknowns satisfy the equality statement. The general form of an equation is ax + by = c. Here, a, and b are non−zero coefficients and c are the constants and x, and y are variables. Two such equations a1x + b1y = c1 and a2x + b2y = c2 are a pair of simultaneous equations in x, and y.

Translating Statement into Mathematical Equations

To solve real-life problems, we need to convert them into mathematical form.

  1. Note down the facts from the problem.
  2. Use Variables for denoting unknown quantities.
  3. Try to establish a relationship between known and unknown quantities.
  4. Form an equation by this relationship.
  5. Solve it to get the desired result.

Methods of Solving a System of Equations in Two Variables

A value of each of the unknown or variable satisfies both the equation simultaneously. These values of the variables are the roots of the equations.

A. Elimination Method

In this method, we eliminate one of the variables (say x) from the equation. In other words, two linear equations are reduced to form only one linear equation by eliminating one of the unknowns. Suppose we have,

x = 16 – y… (a)
and, x = 4 + y … (b)

Since the L.H.S. of both the equations is same. We have, 16 – y = 4 + y or, 12 = 2y or, y = 6 and solving for x, we have x = 10.

B. Cross Multiplication Method

In this method we cross – multiply the equations with respective coefficients. Subtracting the two equations thus formed gives the required answer. If the two equations are

ax + b1 y + c1 = 0, and
a2 x + b2 y + c2 = 0,

in more general form, we have

system of equations

Assume we have,

x + y = 16 … (i)
x – y = 4 … (ii)

Multiplying (i) by (−1) and (ii) by (1), and then subtracting the new equations we have,

−x – x – y + y = −16 – 4

or, −2x = −20

or, x = 10 and solving for y, we have y = 6.

II. System of Equations in Three Variables

A relationship between three variables shown in the form of a system of three equations is a triplet of simultaneous equations. The general form of equations in this form is ax + by + cz = d. Here, a, b, and c are non – zero coefficients, d is a constant. Here, x, y, and z are unknown variables.

Methods of Solving a System of Equations in Three Variables

A value of each of the unknown or variable satisfying all the equation simultaneously gives the roots of the equations.

A. Elimination Method

This method is similar to that in two variables. One of the variables is eliminated to form a linear equation and the equations are then solved. Suppose we have,

2x – y + 3z = 10 …(a)
x + 3y – 2z = 5 …(b)
and, 3x – 2y + 4z = 12 … (c)

Here, we eliminate y by multiplying (a) by 3 and then solving the first two equations. We proceed in a similar way and solving the equations, we have x = 2, y = 3, and z = 3.

B. Cross Multiplication Method

In this method, we find the value of x and y just as we find the values for equations in two variables. From the known values of x and y and using any of the equations we calculate the value of z. Suppose we have,

2x – 2y + 3z = 20 …(a)
x + 3y – z = 12 …(b)
and, 3x – y + 4z = 22 … (c)

Using the first two equations and putting the values in the formula for x and y, we have

x = [(– 2 × (– z – 12)) – 3 × (3z – 20)] ÷ [(2 × 3) – (1 × – 2)]. Here, (3z – 20) and (– z – 12) are the respective c1 and c2.

or, x = (– 7z + 84) ⁄ 8. Similarly, y = (5z + 4) ⁄ 8.

Substituting these values of x, and y in (c) we have

x = 21, y = – 7, and z = – 12.

Solved Example for You

Problem: Which of the following is the correct values of x, and y for the equations

2x – 3y – 12 = 0, and
x – 6y – 15 = 0.

The options are:

  1. x = – 2, y = 3
  2. y = – 2, x = – 3
  3. x = 3, y = – 2
  4. y = – 3, x = – 2

Solution: Solving the above equations by the method of cross – multiplication, we have

x = [(–3 × –15) – (–6 × –12)] ÷ (2 × –6) – (1 × –3)] = 3.

and, similarly, y = – 2. The correct choice is (c) i.e., x = 3 and y = –2

Share with friends

Customize your course in 30 seconds

Which class are you in?
Get ready for all-new Live Classes!
Now learn Live with India's best teachers. Join courses with the best schedule and enjoy fun and interactive classes.
Ashhar Firdausi
IIT Roorkee
Dr. Nazma Shaik
Gaurav Tiwari
Get Started

Leave a Reply

2 Comment threads
0 Thread replies
Most reacted comment
Hottest comment thread
1 Comment authors
आकाश पाटील Recent comment authors
newest oldest most voted
Notify of
आकाश पाटील
आकाश पाटील

सर मी स्पर्धा परीक्षेची तयारी करीत आहे मला महाजनको साठी अभ्यास करायचा आहे

आकाश पाटील
आकाश पाटील

सर मी स्पर्धा परीक्षा ची तयारी करत आहे मला महाजनको साठी अभ्यास करायचा आहे

Stuck with a

Question Mark?

Have a doubt at 3 am? Our experts are available 24x7. Connect with a tutor instantly and get your concepts cleared in less than 3 steps.
toppr Code

chance to win a

study tour

Download the App

Watch lectures, practise questions and take tests on the go.

Get Question Papers of Last 10 Years

Which class are you in?
No thanks.