Solving linear equations

Linear equations are first-order equations. An equation with a homogeneous variable of degree 1 is defined as a linear equation in one variable (i.e. only one variable). There might be more than one variable in a linear equation. If there are two variables in a linear equation, it is referred to as linear equations in two variables, and so on. 

Linear equations

An equation is a mathematical statement that has an equal symbol (=) between the algebraic expressions. When the answers to linear equations are substituted for the unknown values, the equation becomes true. There is just one solution in the case of a single variable. The equation x + 2 = 0 has only one solution, which is x = -2. 

Forms of linear equation

The 3 types of linear equations are simply as the following:

1. Standard form

2. Slope intercept form

3. Point slope form

Standard form of linear equation

Constants and variables are combined in linear equations. A linear equation in one variable is written in standard form as 

ax + b = 0, where, a≠0 and x is the variable

The standard form of a two-variable linear equation is written as 

ax + by + c = 0, where, a≠0, b≠0, x and y are the variables

The standard form of a three-variable linear equation is written as 

ax + by + cz + d = 0, where, a≠0, b≠0, c≠0, x, y, z are the variables

Slope intercept form

The slope-intercept form of linear equations is the most frequent, and it is written as; 

y = mx + b

Where, 

m = slope of the line

b = y-intercept

x and y = coordinates of the x-axis and y-axis respectively.

For example, y = 3x + 7

Slope, m = 3, 7 = Intercept

The x-coordinate will be equal to 0 if a straight line is parallel to the x-axis. Therefore, 

Y=b 

The y-coordinate will be zero if the line is parallel to the y-axis. 

mx + b = 0

x = -b/m

Slope

The ratio of the change in y-coordinates to the change in x-coordinates is the slope of the line.

m = (y₂-y₁)/(x₂-x₁)

So, the slope depicts the ascent of a line in the plane as well as the distance travelled along the x-axis. A gradient is a term used to describe the slope of a line. 

Point slope form

A straight line equation is generated by examining the points in the x-y plane in this form of linear equation, such that: 

y – y₁ = m(x – X₁)

The coordinates of the point are (x₁, y₁).

It can also be written as: 

y = mx + y₁ – mx₁

How to solve linear equations

Solution of linear equations in one variables

Both sides of the equation must be balanced in order to solve a linear equation. Both sides of the ‘equal to’ sign are equal when the equality sign is used. Because the equation is balanced, specific mathematical operations are done on both sides of the equation in such a way that the equation’s balance is preserved. 

Solution of linear equations in two variables

There are various methods for solving two-variable linear equations. Here are a few of them:

1. Substitution method 

2. Method of cross multiplication 

3. Elimination method

To get the values of two variables, we must choose a set of two equations. ax + by + c = 0 and dx + ey + f = 0, also known as a two-variable system of equations, in which x and y are two variables and a, b, c, d, e, f are constants, and a, b, d, and e are not zero. 

Solution of linear equations in three variables

To get the values of unknowns in linear equations in three variables, we require a set of three equations, as shown below. The matrix technique is a popular approach for solving three-variable systems of linear equations. 

A₁x + b₁ y + c₁z + d₁ = 0 

A₂x + b₂y + c₂z + d₂ = 0 and 

A₃x + b₃y + c₃z + d₃ = 0 

Conclusion

Linear equations have a wide range of applications and can be used in a variety of real-life scenarios. We convert the circumstance into mathematical statements to use algebra to deal with real-life issues. So that the relationship between the unknown variables and the known data is plainly seen.