Linear Equations – Standard Forms and Examples

A linear equation or a one-degree equation is one in which the maximum power of the variable is 1. For instance, 4x + y = 6 is a linear equation since both x and y have a maximum power of one. The conventional form of a linear equation is as follows: Ax + By = C, where A, B, and C are integers and the variables are denoted by the letters x and y.

Forms of Linear Equation

Linear equations are classified into three types: 

1. Standard form
2. Slope intercept form
3. Point slope form.

Simple Linear Equations with One Variable in the Standard Form

The term “linear equation in one variable” refers to an equation that has only one variable. This indicates that there is just one solution to this linear equation. The standard form, or general form, of one-variable linear equations, is stated as Ax + B = 0, where A and B are integers and x is the single variable. For instance, 3x + 6 = 12 is the conventional form of a linear equation with a single variable, and when we solve for x, we have exactly one answer, which is 4.

Simple Linear Equations with Two Variables in the Standard Form

When there are two variables in a linear equation, there are two solutions. In two variables, the standard form of linear equations (generic form of linear equations) is stated as,

Ax + By = C

Where A, B, and C are all integers

x and y are variables.

For example, 3x + 4y = 8 is the conventional form of a two-variable linear equation.

Slope Intercept Form

The most frequently seen representation of linear equations is in slope-intercept form, which is denoted by

y = ax + b.

Where,

m denotes the slope of the line, b denotes the y-intercept, and x and y denote the x- and y-axes, respectively.

For example, y = 3x + 9:

slope,

a = 3 and intercept = 9

In this case, if a straight line is parallel to the x-axis, then the x-coordinate will be 0. Therefore,

y=b

If the line is on the same level as the y-axis, then the y-coordinate will be 0.

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. It  can be evaluated by:

m = (y2-y1)/(x2-x1)

So the slope shows the rise of the line in the plane and how far it has gone on the x-axis. The slope of the line is also called a gradient, and it is also called a slope.

Point Slope Form

A straight line equation is created in this form of linear equation by considering the points in the x-y plane such that: y – y1 = m(x – x1), where (x1, y1) is the point’s coordinates.

Additionally, we can write it as y = mx + y1 – mx1.

Cramer’s Rule

Cramer’s Rule is a convenient approach to solve for a single variable in a system of linear equations without having to solve the entire system of equations. They rarely teach Cramer’s Rule in this manner, but this is precisely the point: rather than solving the complete system of equations to obtain the desired value, you can instead apply Cramer’s to obtain the desired value.

Cramer’s Rule Formula

Consider a system of linear equations in the matrix form AX = B with n variables x1, x2, x3,…, xn.

Here, A denotes the matrix of coefficients (must be a square matrix)

X = Variable-column matrix

B = Matrix of columns containing constants (which are on the right side of the equations)

Now we must determine the determinants in the following order: D = |A|, Dx1, Dx2, Dx3,…, Dxn.

Here, Dxi is the same determinant as D for i = 1, 2, 3,…, n, so that the column is substituted by B.

Thus, x1 = Dx1/D; x2 = Dx2/D; x3 = Dx3/D; ….; xn = Dxn/D {where D is not equal to 0}

Gauss–Jordan Method

The Gauss Jordan Elimination, or Gaussian Elimination, is a method for solving a system of linear equations by transforming it into an augmented matrix, lowering it using row operations, and then expressing the problem in the reduced row-echelon form to obtain the values of the variables.

Gaussian Elimination is a method for solving a system of linear equations in an organised manner. As such, it is a programme that can be readily programmed to solve a system of linear equations. Gauss-Jordan Elimination’s primary objective is to: 

• Represent a system of linear equations in augmented matrix form
• Then perform the three-row operations on it until the reduced row echelon form (RREF) is obtained.
• Finally, we can quickly identify the solutions from the RREF.

Consistent and Inconsistent Systems

Consistent System: In mathematics, a consistent system is an equation that has at least one common solution. Consider the following examples of consistent equations: x + y = 6 and x – y = 2. Both of these equations have a common solution. Similarly, the equations x + y = 12 and 3y = x both have a common solution, which is why we can refer to them as consistent equations.

If the equation’s lines intersect or are parallel, a two-variable system of equations is deemed consistent.

Inconsistent System: Inconsistent linear equations are those that lack a common solution. If the equations are graphed on a coordinate plane, the lines in this system will be parallel. Consider the following inconsistency: x – y = 8 and 5x – 5y = 25. They have no agreed-upon remedies.

When the lines or planes generated by the systems of equations do not intersect or are not parallel, an inconsistent system results.

Conclusion

Linear equations are first-order equations. Linear equations are defined in terms of coordinate lines. When an equation contains only one homogeneous variable of degree one (i.e., one variable), it is referred to as a linear equation in one variable. A linear equation may contain multiple variables. If a linear equation contains two variables, it is referred to as a linear equation with two variables, and so on.