LINEAR EQUATIONS

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

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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