Introduction To Calculus

Linear Equations in One Variable

 

23 September 2020

Reading Time: 5 Minutes

“Algebra is a general method of computation by certain signs and symbols which have been contrived for the purpose and found convenient”

Algebra is a branch of Mathematics in which letters or alphabets and other general symbols are used to represent numbers and quantities in different formulae and equations.

Algebra makes it easy to establish mathematical relationships. Algebra deals with variables, constants and arithmetic operations.


What is Algebraic Equation?

Algebraic expression is an expression which is made up of constants, variables and algebraic operations. Algebraic equation is a statement which states that two algebraic expressions are equal.
Example: \(3x+1=7, 4x+1=x+4\)


What is Linear Equation?

A Linear Equation is an equation for a straight line. A linear equation is a algebraic equation where each term has an exponent one.

The linear equation can broadly classify into two types. They are:

  1. Linear Equation in One Variable

The equation involving only one variable in first order is called Linear Equation in one variable.

Example: \(2x+1=3x+5, 9x+2=18\)

The standard form of Linear Equation is Ax+By=C; where, x and y are the variables, C is Constant and A and B are the coefficient of   x and y respectively.

  1. Linear Equation in Two Variables

The equation involving two variables and their degree is 1 is called a Linear Equation in two variables.

Example:  \(2x+3y=5 , 3a+7b=80\) 


Solution of a Linear Equation

A solution of a linear equation is the assignment of the values of  variable x1, x2, ...,so that each of the equations is satisfied, which means the Left Hand Side (LHS) is equal to the Right Hand Side (RHS).

The solution of the linear equation is infinite. The set is a collection of well defined and distinct objects. The collection of possible solutions is called a solution set.


Example Problems

1. The sum of two numbers is 84. If one number is six times another number, find the numbers.

Answer: Let the first number be “\(x\)”.

Let the second number be 6x

The equation is \(x+6x=84\)

                           \(7x=84\)
                            \(x=84/7\)
                            \(x=12\)

The first number is \(x=12\)

The second number is \(6x=6*12=72\)

The numbers are 12 and 72

2. A total Rs. 51,000 is distributed to the students as price money for a competition. The cash contains Rs. 100 notes and Rs. 500 notes in the ratio 2:3. Find the total number of Rs. 100 notes and Rs. 500 notes

Answer: Let the common ratio of number of notes be “x”

Number of Rs. 100 notes = \(2x\)

Number of Rs. 500 notes = \(3x\)

Total Amount = \(2x*100+3x*500\)
        \( 51, 000 = 200x+1500x\)
        \( 51, 000 =1700x\)
                 \( x=30\)

Therefore, Number of Rs. 100 notes are \(2x=2*30=60\)

Number of Rs. 500 notes are \(3x=3*30=90\)

Rs.100 notes=60 and Rs. 500 notes=90

3. Find \(x\): \(3(x-5) +2(x-2) =4+7(x-4)\) 

Answer:  \(3(x-5) +2(x-2) =4+7(x-4)\)
                   \(3x-15 + 2x -4 = 4 + 7x – 28\)
                                  \(5x-19=7x-24\)
                                  \( 5x-7x=-24+19\)
                                        \( -2x=-5\)
                                            \(x=5/2\)

\(x=5/2\)

4. If \(1.7y + 2.3y= 2\), then \(y\) is __________________
(A) \(1/4\)  
(B)  \(1/2\)
(C) 8
(D) 6

Answer: \(1.7y + 2.3y= 2\)
                            \(4y=2\)
                              \(y = 1/2\)

\(y = 1/2\)

5. Three chairs and four tables cost Rs. 2800. If the price of each chair is Rs. 600, what is the price of each table (in Rs)?
(A) 250
(B) 350
(C) 100
(D) 150

Answer: Let the chair be denoted as “c”

Let the table be denoted as “t”

\(3c+4t = 2800\)

\(3*600+4t = 2800\)

\(1800 + 4t = 2800\)

\(4t=2800-1800\)

\(t=1000/4=250\)

The price of each table is (A) Rs. 250

6. If seven times a number is added to one-fifth of itself, then five-sixth of the sum is equal to 30. Find the number.

(a) 5
(b) 6
(c) 15
(d) 10

Answer: Let the number be “\(x\)

Linear Equation = \(5/6(7x+1/5x) =30\)
                                   \(5/6*36x/5 = 30\)
                                       \(180x/30= 30\)
                                            \(180x=900\)
                                                  \(x=5\)

\(x=5\)


Learning Outcomes

  • Finding the values of variables for which LHS and RHS becomes equal, means LHS=RHS
  • Understanding how the equation has to be solved step by step
  • How to frame equations by carefully observing and understanding the words.

Conclusion

I hope the blog has helped the students to understand the linear equations of single variables and steps to solve them.

The example problems will help in the better understanding of the topic in the practical environment. Solving linear equations becomes easy when you approach the equation step by step and apply the appropriate operations.

Cuemath, a student-friendly mathematics platform, conducts regular Online Live Classes for academics and skill-development, providing students with real-time feedback, instant doubt clearing, assignments for homework, and much more.

Their Mental Math App, on both iOS and Android, is a one-stop solution for kids to develop multiple skills like typing, critical thinking, decision-making, problem-solving, etc. through games, puzzles, quizzes, simulations, and so on.


Frequently Asked Questions (FAQs)

What is linear equation formula?

The Linear equation formula is Ax+By=C; where, x and y are the variables,
A and B are coefficient of x and y respectively,
C is constant.

What are the steps to solve linear equations?

Step 1: The main objective to find the solution of the variable. So, simplify each side separately by applying the operations  (Add/Subtract/Multiply/Divide)
Step 2: Transfer all the knowns to one side and unknowns to another side.
Step 3: Find the value of the unknown

What is linear and non-linear equation?

Linear equation is the equation of a straight line, whereas Non-linear equation is an equation of a curve.

-Written by Anusha Makam D R, Cuemath Teacher

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