Linear equations are those equations that are of first order. Linear equations are also first-degree equations as it has highest exponent of variables as 1.Â Some of the examples of linear equations are as follows:
- 2x – 3 = 0,Â
- 2y = 8
- Â m + 1 = 0,
- Â x/2 = 3
- Â x + y = 2
- 3x – y + z = 3
When the equation has a homogeneous variable (i.e. only one variable), then this type of equation is known as a Linear equation in one variable.
Linear Equation Definition
What is a linear equation definition and example? An Equation having the maximum order of 1 is known as a Linear equation.
Below are some examples of linear equations in 1 variable, 2 variables and 3 variables:
Linear Equation in One variable | Linear Equation in Two variable | Linear Equation in Three variable |
3x+5=0
\(\frac{3}{2}x +7 = 0\)Â 98x = 49 |
y = 7x=3
3a+2b = 5 6x+9y-12=0 |
x + y + z = 0
a – 3b = c 3x + 12 y = Â½ z |
There are different forms to write linear equations. Some of them areÂ
Linear Equation | General Form | Example |
Slope intercept form | y = mx + c | y + 2x = 3 |
Pointâ€“slope form | y – y_1 = m(x – x_1) | y – 3 = 6(x – 2) |
General Form | Ax + By + C = 0 | 2x + 3y – 6 = 0 |
Intercept form | x/x_0 + y/y_0 = 1 | x/2 + y/3 = 1 |
As a Function | f(x) instead of y
f(x) = x + C |
f(x) = x + 3 |
The Identity Function | f(x) = x | f(x) = 3x |
Constant Functions | f(x) = C | f(x) = 6 |
Where m = slope of a line; (x_0, y_0) intercept of x-axis and y-axis.
Standard FormÂ
Linear equations are the combination of constants and variables.Â
The standard form of a linear equation in one variable is represented as
where, a â‰ 0, b â‰ 0 , x and y are the variables.
The standard form of a linear equation in two variables is represented as
ax + byÂ + c = 0, where, a â‰ 0, b â‰ 0 , x and y are the variables. |
The standard form of a linear equation in three variables is represented as
ax + by + cz + d = 0 where a â‰ 0, b â‰ 0, c â‰ 0, x, y, z are the variables. |
How to Solve Linear Equations
Solving Linear Equations in One Variable
Both sides of the equation are supposed to be balanced for solving a linear equation. Equality sign denotes that the expressions on either side of the â€˜equal toâ€™ sign are equal. Since the equation is balanced, for solving it certain mathematical operations are performed on both sides of the equation in a manner that it does not affect the balance of the equation. [Use BODMAS Rule]
For Example:
2x – 10/2 = 3(x – 1)
Step 1: Clear the fraction
2x – 5 = 3(x – 1)
Step 2: Simplify Both sides equations
Â 2x – 5 = 3x – 3
2x = 3x + 2
2x – 3x = 2
Step 3: Isolate x
x = -2
Solving Linear Equations in Two Variables
Solving Linear Equations having 2 variables, In mathematics, there are different methods. Following are some of them:
- Method of substitution
- Cross multiplication method
- Method of elimination
- Determinant methods
We must choose set of 2 equations to find the values of 2 variables. Such as ax + by + c = 0 and dx + ey + f = 0, also called a system of equations with two variables, where x and y are two variables and a, b, c, d, e, f are constants, and a, b, d and e are not be zero. Else single equation has an infinite number of solutions.
Solving Linear Equations in Three Variables
To solve Linear Equations having 3 variables, we need a set of 3 equations as given below to find the values of unknowns. Matrix method is one of the popular methods to solve system of linear equations with 3 variables.
a_{1}x + b_{1} y + c_{1}z + d_{1} = 0
a_{2}x + b_{2} y + c_{2} z + d_{2 }= 0 and
a_{3}x + b_{3} y + c_{3} z + d_{3 }= 0
Learn more on the related Topics:
- Application of linear equations
- Linear Equations in One variableÂ
- Linear Equations Two Variables
- Graphing Of Linear Equations
- Linear Equations In Two Variables Class 9
Linear Equations Example
Example 1: Solve xÂ = 12(x +2)
Solution:Â
xÂ = 12(xÂ + 2)
x = 12x + 24
Subtract 24 from each side
x – 24 = 12x + 24 – 24
x – 24 = 12x
Simplify
11xÂ = -24
Isolate x, by dividing each side by 11
11x / 11 = -24/11
x = -24/11
Example 2: Solve x – y = 12 and 2x + y = 22
Solution:Â
Name the equations
x – y = 12Â Â Â Â Â Â Â Â Â Â Â Â ———- (1)
2x + y = 22 Â Â Â Â Â Â Â Â Â Â Â ———- (2)
Isolate Equation (1) for x,
x = y + 12
Substitute y + 12 for x in equation (2)
2(y+12) + y = 22
3y + 24 = 22
3y = -2
or y = -2/3
Substitute the value of y in x = y + 12
x = y + 12
x = -2/3 + 12
x = 34/3
Answer: x = 34/3 and y = -2/3