Worried about how to solve linear equations? Then, don’t panic anymore as we have come up with an article that assists you in learning the concept of the linear equation well. Be well versed with the details like what is meant by a linear equation, different forms of linear equations, some of the key features of linear equations. In fact, we have provided detailed steps on how to solve linear equations manually along with worked-out examples for better grip on the concept.
What is a Linear Equation?
An equation is defined as a combination of variables and/or numbers. Similarly, a linear equation is constructed from two expression sets, that are equal to each other. The first order of equations is defined as Linear equations. The linear equations are determined for lines in the coordinate system. When the equation has one variable with degree 1, then it is known as a linear equation in one variable. Whereas, the linear equation can have more than one variable. So, the linear equation with two variables is called linear equations in two variables.
For more clarity, the two variables linear equation is nothing but the relationship between x and y; the value of one variable from the two variables i.e mostly y, depends on the value of the other i.e mostly x. Here, x is the independent variable, and y is called as the dependent variable.
For linear equation in n variables forms a0 + (a1x1) + … + (anxn) = c, here x1.. xn are variables, a1..a0 are coefficients, a1… an and c is a constant. The equation might contain more variables, in this case, the differentiation of equations is done on a certain basis; in some cases equation can be linear depending on the variable. Like, the equation x + y = 6 is linear in both x and y, but in x + y6 = 0 equation is linear in x but not in y.
Key Features of Linear Equations
- The most common and simplest thing to remember about the linear equation is; it has only one or two variables.
- None of the variables in a linear equation has a power greater than 1, nor is used as the denominator of a fraction.
- The equation includes = sign and it is true on both LHS and RHS sides.
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Linear Equation Forms
There are three major forms of linear equations that are as follows:
- Standard Form
- Slope Intercept Form
- Point Slope Form
Standard Form of Linear Equation
The standard form of linear equation is with a combination of constants and variables.
This type of equation has one variable and that is represented as
ax + b = 0, here, a ≠0 and x = variable.
Two variables linear equation is:
ax + by + c = 0, here, a ≠0, b ≠0 , x and y are the variables.
Three variables standard form linear equation is:
ax + by + cz + d = 0, here a ≠0, b ≠0, c ≠0, x, y, z are the variables.
Slope Intercept Form
Slope-intercept linear equation is represented as: y = mx + b
Here, m is the slope of the line, x and y are the coordinates of the x-axis and y-axis and b is the y-intercept.
Slope: The slope of a line is also known as a gradient. The ratio of change in y-coordinates to the change in x-coordinates is equal to the slope of the line.
Represntation of slope is done as following:
m = (y2-y1)/(x2-x1)
Point Slope Form
In Point slope form, the straight-line equation is formed by evaluating the points in the x-y plane, such that:
y – y1 = m(x – x1 )
Here, (x1, y1) are the coordinates of the point.
It can be also represented as :
y = mx + y1 – mx1
Linear Equation Formula
Linear equation formula is nothing but the way of expressing a linear equation. There are different ways to represent the linear equation formula. As, per the forms of linear equation, in the standard form, it is clear that it varies in every case as it is based on the number of variables and the most common thing to remember is that there is only one and the highest degree of all variables in the equation that is 1.
How to Solve Linear Equations?
Follow the simple steps listed below and learn how to solve linear equations manually by hand. The step by step process for solving linear equations is as follows
- Both sides of the equations must be equal. Like, if we do any kind of formulation (addition, subtraction, multiplication, and division) on both sides of the equation it is maintained as true only.
- The first basic step of the linear equation solution is to bring the variables on the one side of the equation, constant to the other side, and then find the value of the unknown variable.
Check out the following example for a better understanding of solving the linear equations:
Example:
Solve the equation, 4x-8=8
Solution:
Perform the mathematical operations on both the Left-hand side (LHS) and the right-hand side (RHS) to ensure that balance is not disturbed. So, let’s add 8 on both sides.
Without disturbing the balance from both LHS and RHS. We get the new LHS that is 4x-8+8 that gives 4x, new RHS is 4+8 gives 12. Now, after dividing both sides by 4 LHS will be x. In short we get x = 3.
This is the simplest way of solving linear equations that has one variable.