Linear Equations in one variable

Linear Equations in one variable
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Linear equation in one variable is the basic equation used to represent and solve for unknown quantities. The linear equation in one variable can be easily represented graphically and it is always a straight line. The linear equation is an easy way of representing a math statement. Any variable or symbol can be used to represent unknown quantities but generally, a variable 'x' is used to represent the unknown quantity in the linear equation in one variable. 

Solving a linear equation includes a set of simple methods. The variables are isolated on one side of the equation and the constants are isolated to the another side of the equation, to obtain the final value of the quantity. Let us learn more about linear equations in one variable and the methods to solve in this lesson.

Table of Content

What are Linear Equations in One Variable?

Before learning about linear equations in one variable, let us quickly go through the meaning of linear equations. A linear equation is a type of equation in which the degree of each variable in the equation is exactly equal to one. When it is drawn on a graph, it appears to be a straight line either horizontally or vertically. Linear equations in one variable are those equations in which there is only one variable present, and there is only one solution of the equation.

Graph of a Linear Equation

A linear equation in one variable is of the form of ax+b=0, where a and b are any two integers and x is an unknown variable having only one solution. Let us understand this by taking an example. "4 added to a certain number gives 10". Find that number. How can we represent this problem in a simpler way? We can say, x + 4 = 10 , find x. We assigned a variable to that number; this is called an equation. It helps us write large problems like this in a shorter way. This equation has one variable, which is x and the highest power of x is one. These kinds of equations are known as linear equations in one variable because the degree of the variable x is one.

How to Solve Linear Equations in One Variable?

The general form of a linear equation in one variable is Ax + B = 0. Here A is the coefficient of x, x is the variable, and B is the constant term. The coefficient and the constant term should be segregated to find the final solution of this linear equation.

Terms in a Linear Equation

Now, let us look at how to solve a one-variable linear equation. An equation is like a weighing balance with equal weights on both sides.

Balanching a Linear Equation

If we add or subtract the same number from both sides of an equation, it still holds. Similarly, if we multiply or divide the same number on both sides of an equation, it still holds. Consider the equation, 3x-2=4. We will perform mathematical operations on the LHS and the RHS so that the balance is not disturbed. Let's add 2 on both sides to reduce the LHS to 3x. This will not disturb the balance. The new LHS is 3x-2+2=3x and the new RHS is 4+2=6.Now let's divide both sides by 3 to reduce the LHS to x. Thus, we have 3x/3=6/3. Hence we have x = 2.

The above steps to solve linear equations in one variable can be summarized in the below-listed points.

  • Keep the variable term on one side and constants on another side of the equation by adding or subtracting on both sides of the equation.
  • Simplify the constant terms.
  • Isolate the variable on one side by multiply or dividing it into both sides of the equation.
  • Simplify and write the answer.

Linear Equations in One Variable vs Non-Linear Equations

Apart from linear equations in one variable, we have other non-linear equations, which have numerous applications in geometry, trigonometry, calculus. Linear equations in one variable are a single-degree equation and are represented as a line in coordinate axes. A non-linear equation on the other hand is a curve or one linear representation in the coordinate axis. A nonlinear equation is of a higher degree. An example of a non-linear equation is the equation of curve such as a circle, parabola, ellipsehyperbola

Some of the samples of linear equations are x = 5, 3x + 7 = 9, 4x + 2y = 11. And some of the examples of non-linear equations are the equation of a circle - x2 + y2= 25, equation of a elipse - x2/9 + y2/16= 1, equation of a hyperbola - x2 /16 - y2/ 25 =1. 

Important Notes

The following points help us in clearly summarizing the concepts involved in linear equations in one variable.

  1. The degree of the variable in linear equations should be exactly equal to one.
  2. The graph of a linear equation in one variable is a straight line.
  3. The solution of a linear equation is unaffected if any number is added, subtracted, multiplied, or divided on both sides of the equation.

Linear Equations in One Variable Related Topics

Given below is the list of topics that are closely connected to linear equations in one variable. These topics will also give you a glimpse of how such concepts are covered in Cuemath.


FAQs on Linear Equations in One Variable

What are the Types of Linear Equations?

Types of linear equations are classified on the basis of a number of variables in the equation. Majorly, there are three types of linear equations:

  • Linear equation in one variable.
  • Linear equation in two variables.
  • Linear equation in three variables.

What are the Three Types of Solutions of Linear Equations?

Three types of solutions of linear equations are given as:

  • One unique solution.
  • No solution.
  • Infinite number of solutions.

What is a Linear Equation in One Variable with Example?

Linear equation in one variable is of the form ax + b = 0. The linear equation in one variable are equations in which the highest degree of every term is one, there is one possible solution of the equation and there is only one variable present in it. An example of a linear equation in one variable is, 3y+2=0.

What is the Highest Power of Linear Equation?

A linear equation has the highest power of 1. Referring to an example of a linear equation, 3x + 4 = 11, the power of the variable 'x' is 1.

Can a Linear Equation Have More than One Variable?

Yes, linear equations can have more than one variable. We call such equations as linear equations in two variables or linear equations in three variables. The linear equations in two variables are of the form ax + by + c = 0 and a linear equation in three variables is of the form ax + by + cz + d = 0. Here x, y, z are the variables, and a, b and c are coefficients and d is a constant.

How do you Solve Linear Equations with Only One Variable?

Steps to solve linear equations in one variable are listed below:

  • Keep the variable term on one side and constants to another side of the equation by adding or subtracting on both sides of the equation.
  • Simplify the constant terms.
  • Isolate the variable on one side by multiply or dividing on both sides of the equation.
  • Simplify and write the answer.

How do you Solve Linear Equations With Variables on Both Sides?

To solve linear equations with variables on both sides, we first bring all the terms with variables on one side and the constants on the other side of the equation. Then, we simplify the equation, isolate the variable and write the final answer of the equation. Let us look at a simple equation to understand this. 4x + 1 = 2x + 7; 4x - 2x = 7 - 1; 2x = 6; x = 6/2; x = 3.

What are the Rules for Solving Linear Equations?

The most general rule for solving linear equations is that we can add, subtract, multiply or divide the same term on both sides of the equations so that we can find the value of the variable present in it.

How do you know if a System is Linear or Dependent?

An equation is linear when there is only one variable present in it, while in dependent equations, there is a minimum of two variables present and the value of one is dependent on the value of other variables.


Solved Examples

Example 1: Twenty years ago, Mikkel's age was one-third of what it is now. What is Mikkel's present age?

Solution:

Let Mikkel's present age be = x years. Twenty years ago, Mikkel's age was = x-20 years. According to the given information, x - 20= x/3; 3(x - 20)= x ; 3x - 60= x ; 3x - x= 60 ; 2x= 60 ;  x= 60/2 ; x= 30 . Therefore the present age of Mikkel is 30 years.

Example 2: David worked as a stenographer. In June, he was paid $50 per day. However, $10 per day was deducted for the days he remained absent. He received $900 for the number of days he worked. How many days did he work?

Solution:

Let the number of days he worked be = x. Hence, the number of days he did not work will be = 30 - x. He was paid $50 for each day he worked and $10 was deducted for each day he did not work. At the end of the month, he received $900. According to the given information, 50(x) - 10(30 - x) = 900;  50x - (300 - 10x) = 900;  50x - 300 + 10x  = 900;  60x= 900 + 300 ;  x = 1200/60;  x= 20. Therefore David worked for 20 days.

Example 3: Find the perimeter of the square whose side length, x is given in the form of an equation as 2x/3- 5/6=0

Solution:

Given, side-length of the square is 2x/3- 5/6=0.  First, we need to solve this equation to find the value of x. 2x/3- 5/6=0; 2x/3- 5/6 + 5/6=0+ 5/6; 2x/3= 5/6; x=5/6 × 3/2; x= 5/4; Hence, the side-length of the square is 5/4 units. Now, to find the perimeter of the square, we need to multiply the side-length by 4. The perimeter of a square = 5/4 × 4=5 units. Therefore the perimeter of the square is 5 units.

Practice Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

 
 
 
 

 

 

  
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