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 = z^{2}
If both the variables x and y are equal to the variable z, then xy becomes z^{2}. 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
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 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
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
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
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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