Linear equations are first-order equations, which implies that the highest power of the variable in these equations is 1. For example, x + 2 = 20, y – 33 = x, etc. Linear in itself means residing in the same line. These equations apply to lines in a coordinate system. A linear equation is usually the equation of a straight line. The straight-line equation is generally represented as y = mx + b, where m is the slope or projection of the line in the cartesian plane and b is the y-intercept. These equations apply to lines in a coordinate system.
In one-variable: When an equation has a homogeneous variable (i.e., only one variable), it is referred to as a Linear equation in one variable. In other terms, a line equation is derived by connecting zero to a linear polynomial over any field and obtaining the coefficients.
In two-variable: When a linear equation has more than one variable (specifically two), it is termed a linear equation in two variables. The equation of any straight-line y = kx + b is the prime example of a linear equation in 2 variables.
Linear equation solutions provide values that, when substituted for the unknown variables, make the equation true. There is only one solution in the case of a single variable, such as x + 2 = 0. However, in the case of a two-variable linear equation, the solutions are computed as the Cartesian coordinates of a point on the Euclidean plane.
When solving a linear equation, both sides of the equation must be balanced. The equality sign indicates that the phrases on each side of the 'equal to' sign are equivalent. Because the equation is balanced, specific mathematical operations like addition, multiplication, etc., can be done on both sides of the equation to solve it in a way that does not compromise the equation's balance. The following are the general steps for solving linear equations:
Find the solution, i.e., the value of x and note it down. From the above equation 23x – 13x = -20 – 10, 10x = -30, therefore x = -3.
Substitute this value of x back into the original equation to verify the answer.
23 * (-3) + 10 = 13 * (-3) – 20
-69 + 10 = -39 – 20
-59 = -59
Left-hand side = Right-hand side