Linear Equations

Linear equations are equations of the first order. These equations are defined for lines in the coordinate system. An equation for a straight line is called a linear equation. The general representation of the straight-line equation is y=mx+b, where m is the slope of the line and b is the y-intercept.

Linear equations are those equations that are of the first order. These equations are defined for lines in the coordinate system. 

Linear equations are also first-degree equations as it has the highest exponent of variables as 1.  

Examples:

• 2x – 3 = 0, 
• 2y = 8
•  m + 1 = 0,
•  x/2 = 3
•  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. In different words, a line equation is achieved by relating zero to a linear polynomial over any field, from which the coefficients are obtained. 

The solutions of linear equations will generate values, which when substituted for the unknown values, make the equation true. In the case of one variable, there is only one solution, such as x+2=0. But in case of the two-variable linear equation, the solutions are calculated as the Cartesian coordinates of a point of the Euclidean plane.

Definition

What is a linear equation definition and give examples? 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:

Equation of a Line

The equation of a straight line is given by:

y = mx + b

Where m is the slope of the line,

b is the y-intercept 

x and y are the coordinates of x-axis and y-axis, respectively.

If a straight line is parallel to x-axis, then x-coordinate will be equal to zero. Therefore,

y=b

If the line is parallel to y-axis then y-coordinate will be zero.

mx+b = 0

x=-b/m

Slope: Slope of the line is equal to the ratio of change in y-coordinates and change in x-coordinates. It  can be evaluated by:

m = (y₂-y₁)/(x₂-x₁)

So basically the slope shows the rise of line in the plane along with the distance covered in x-axis. Slope of line is also called a gradient.

Formulas

There are different forms to write linear equations. Some of them are:

Where m = slope of a line; (x₀, y₀) intercept of x-axis and y-axis.

Also, read:

Forms of Linear Equation

There are many forms through which a line is defined in an X-Y plane. Some of the common forms used here for solving linear equations are:

• General Form
• Slope Intercept Form
• Point Form
• Intercept Form
• Two-Point form

Standard Form of Linear Equation

Linear equations are a combination of constants and variables. 
The standard form of a linear equation in one variable is represented as ax + b = 0 where, a ≠ 0 and x is the variable.
The standard form of a linear equation in two variables is represented as

The standard form of a linear equation in three variables is represented as

Slope Intercept Form

The most common form of linear equations is in slope-intercept form, which is represented as;

y = mx + c

where y and x are the point in x-y plane, m is the slope of the line (also called gradient) and c is the intercept (a constant value).

For example, y = 3x + 7:

slope, m = 3 and intercept = 7

Point Slope Form

In this form of linear equation, a straight line equation is formed by considering the points in x-y plane, such that:

y – y₁ = m(x – x₁ )

where (x₁, y₁) are the coordinates of the line.

We can also express it as:

y = mx + y₁ – mx₁

Intercept Form

A line which is neither parallel to x-axis or y-axis nor it pass through the origin but intersects the axes in two different points, represents the intercept form. The intercept values x₀ and y₀ of these two points are nonzero and forms an equation of the line as:

x/x₀ + y/y₀ = 1

Two-Point Form

If there are two points say, (x₁, y₁) and (x₂, y₂) and only one line passes through them, then the equation of the line is given by:

y – y₁ = [(y₂ – y₁)/(x₂ – x₁)](x – x₁ )

where (y₂ – y₁)/(x₂ – x₁) is the slope of the line and x₁ ≠ x₂

How to Solve Linear Equations

By now you have got an idea of linear equations and its different forms. Now let us learn how to solve linear or line equations in one variable, in two variables and in three variables with examples. Solving these equations with step by step procedures are given here.

Solution of 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 ‘equal to’ 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. Here is the example related to the linear equation in one variable.

Example: Solve (2x – 10)/2 = 3(x – 1)

Step 1: Clear the fraction

x – 5 = 3(x – 1)

Step 2: Simplify Both sides equations

 2x – 5 = 3x – 3

x = 3x + 2

x – 3x = 2

Step 3: Isolate x

-2x = 2
x = -1

Solution of Linear Equations in Two Variables

To solve Linear Equations having 2 variables, there are different methods. Following are some of them:

1. Method of substitution
2. Cross multiplication method
3. Method of elimination
4. Determinant methods

We must choose a 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 zero. Else, the single equation has an infinite number of solutions.

Solution of 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.

a₁x + b₁ y + c₁z + d₁ = 0

a₂x + b₂ y + c₂ z + d₂ = 0 and

a₃x + b₃ y + c₃ z + d₃ = 0
Also check: Solve The Linear Equation In Two Or Three Variables

Problems and Solutions

Example 1: Solve x  = 12(x +2)

Solution: 

x  = 12(x  + 2)

x = 12x + 24

Subtract 24 from each side

x – 24 = 12x + 24 – 24

x – 24 = 12x

Simplify

11x  = -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: 

Name the equations

x – y = 12                        ———- (1)

2x + y = 22                       ———- (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

Practice Questions

Solve the following linear equations:

1. 5y-11=3y+9
2. 3x + 4 = 7 – 2x
3. 9 – 2(y – 5) = y + 10
4. 5(x – 1) = 3(2x – 5) – (1 – 3x)
5. 2(y – 1) – 6y = 10 – 2(y – 4)
6. y/3 – (y – 2)/2 = 7/3
7. (y – 3)/4 + (y – 1)/5 – (y – 2)/3 = 1
8. (3x – 2)/3 + (2x + 3)/3 = (x + 7)/6
9. (8y – 5)/(7y + 1) = -4/5
10. (5 – 7y)/(2 + 4y) = -8/7

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