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.
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.
The 3 types of linear equations are simply as the following:
Standard form
Slope intercept form
Point slope form
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
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
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.
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
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.
There are various methods for solving two-variable linear equations. Here are a few of them:
Substitution method
Method of cross multiplication
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.
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
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.
Profile
Settings
Refer your friends
Sign out
