LINEAR EQUATIONS

Linear equation is an equation which contains variable or variables with the maximum exponent of 1 and minimum exponent of 0.

For example, consider the following equation.

3x + 5 = 0

In the above equation, the term 3x contains the variable x with exponent 1 and the term 5 does not contain the variable. So, the exponent of the variable in the term 5 is zero.

Since the exponents of the variable in 3x + 5 = 0 are 1 and 0, the equation 3x + 5 = 0 is a linear equation. Since we find only one variable x in this equation, it is a linear equation in one variable.

Is the following equation linear ?

xy + 3 = 0

The above equation is not linear. because x and y are multiplied. According to the product rule of exponents, if two same variables are multiplied, the exponents have to be added.

In the equation equation above, if x = y = z, then

xy = zz = z2

If both the variables x and y are equal to the variable z, then xy becomes z2. So, we can consider the exponent of xy as 2. Since one of the terms has the variable with exponent 2, it is not a linear equation.

Is the following equation linear ?

x + 1/x = 2

The above equation is not linear. 

We can rewrite the given equation as follows.

x + x-1 = 2

Since one of the terms has the variable with exponent -1, it is not a linear equation.

Examples of linear equations with two variables :

y = 3x - 5

2x - 3y = 2

Examples of linear equations with three variables :

3x + y - 5z = -1

y = 3x - 2z

Graphs of Linear Equations

Graph of a linear equation in one variable or two variables will always be a straight line.

Graph of a linear equation in three variables will always be a plane.

Solving a Linear Equation

Solving a linear equation is to find the value of the variable or variables the equation contains.

Simple Equation :

A linear equation with only one variable is called simple equation.

To solve a simple equation, only one equation is enough.

System of Linear Equations in Two variables. :

If a linear equation contains two variables and you want to find the values of both variables, then you must have two equations.

A pair of linear equations which contain the same set of two variables is called a system of two linear equations in two variables.

Example :

3x + 2y = 5

2x y = 1

System of Linear Equations in Three variables. :

If a linear equation contains three variables and you want to find the values of all the three variables, then you must have three equations.

A set of three linear equations which contain the same set of three variables is called a system of two linear equations in three variables.

Example :

x + y - z = 1

3x + 2y - z = 4

2x - 3y + 4z = 3

Solving a Linear Equation in One Variable

To solve a linear equation in one variable, you have to isolate the variable. That is, you have to get rid of all the values around the variable.

Example 1 :

Solve :

3x + 5 = 11

Solution :

3x + 5 = 11

Subtract 5 from both sides.

3x = 6

Divide both sides by 3.

x = 2

Example 2 :

Solve :

5(x + 3) - 2(x + 4) = 2x + 1

Solution :

5(x + 3) - 2(x + 4) = 2x + 1

Use Distributive Property.

5x + 15 - 2x - 8 = 2x + 1

3x + 7 = 2x + 1

Subtract 2x from both sides.

x + 7 = 1

Subtract 7 from both sides.

x = -6

Solving a System of Linear Equations in Two Variables

To solve a system of linear equations in two variables, one of the following methods can be used.

(i) Elimination Method

(ii) Substitution Method

(iii) Cross Multiplication Method

Example 3 :

Solve the following system of linear equations using elimination method.

3x + 2y = 8

7x - 2y = 12

Solution :

3x + 2y = 8 ----(1)

7x - 2y = 12 ----(2)

In the above two equations, the coefficient of y-terms are same with different signs.

By adding those two equations, you can eliminate y-terms.

(1) + (2) :

10x = 20

Divide both sides by 10.

x = 2

Substitute x = 2 in (1).

3(2) + 2y = 8

6 + 2y = 8

Subtract 6 from both sides.

2y = 2

Divide both sides by 2.

y = 1

Therefore, the solutions for the given system of linear equations are

x = 2 and y = 1

Example 4 :

Solve the following system of linear equations substitution method.

y = 7x - 3

x + 3y = 13

Solution :

y = 7x - 3 ----(1)

x + 3y = 13 ----(2)

Substitute y = 7x - 3 in (1).

x + 3(7x - 3) = 13

Use Distributive Property.

x + 21x - 9 = 13

22x - 9 = 13

Add 9 to both sides.

22x = 22

Divide both sides by 22.

x = 1

Substitute x = 1 in (1).

y = 7(1) - 3

y = 7 - 3

y = 4

Therefore, the solutions for the given system of linear equations are

x = 1 and y = 4

To learn how to solve a system of linear equations with two variables, please click here

Solving Word Problems Using Linear Equations

Example 5 :

The denominator of a fraction exceeds the numerator by 5. If 3 be added to both, the fraction becomes 3/4. Find the fraction.

Solution :

Let x be the numerator.

Since the denominator of the fraction exceeds the numerator by 5, the fraction is 

= x/(x + 5) ----(1)

Given : If 3 be added to both, the fraction becomes 3/4.

From the above information, we have

(x + 3)/(x + 5 + 3) = 3/4

Simplify.

(x + 3)/(x + 8) = 3/4

4(x + 3) = 3(x + 8)

4x + 12 = 3x + 24

x = 12

Substitute x = 12 in (1).

fraction = 12/(12 + 5)

= 12/17

So, the required fraction is 12/17.

Example 6 :

The total number of students in a school is 501. If the number of boys is equal to 3 more than twice the number of girls, find the number of boys and girls. 

Solution :

Let b be the number of boys and g be the number of girls in the school.

From the given information,

b + g = 501 ----(1)

b = 2g + 3 ----(2)

We can solve the above system of linear equations in two variables using substitution method.

Substitute b = 2g + 3 in (1).

2g + 3 + g = 501

3g + 3 = 501

Subtract 3 from both sides.

3g = 498

Divide both sides by 3.

g = 166

Substitute g = 166 in (2).

b = 2(166) + 3

b = 332 + 3

b = 335

Therefore,

number of boys = 335

number of girls = 166

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