Solution of a Linear Equation in One Variable

As discussed in previous topic of this unit, linear equation is a mathematical statement or equation that has only one variable in it. We know that to solve variables in equation number of equation should be equal to the number of the variables. So, to solve variable present in a linear equation of one variable, one equation is enough to solve the variable.

Below are given some examples of linear equation in one variable:

1. 2x + 3 = 35

2. 3y + 34 = 8

3. 2z +15 = 89

4. 18x +45 = 23

Above are the examples of linear equations in one variable.

Now following are the steps used in solving a linear equation in one variable:

Step I: Observe the linear equation carefully.

Step II: Carefully note the quantity you need to find out.

Step III: Divide the equation in two parts, i.e., L.H.S. and R.H.S.

Step IV: Figure out the terms containing constants and variables.

Step V: Transfer all the constants on the Right Hand Side (R.H.S) of the equation and variables on the Left Hand Side (L.H.S.) of the equation.

Step VI: Perform the algebraic operations on both sides of the equation to get the value of the variable.


Let us solve few examples to understand the concept in a better way.

1. Solve x +12 = 23.

Solution:

Let us first transfer the constants and variables on the R.H.S. and L.H.S. respectively. So,

   x = 23 - 12

   x = 11.

So, value of ‘x’ is 11.


2. Solve 2x +13 = 43.

Solution:

Transfer the constants and variables on their respective sides. So,

2x = 43 - 13

2x = 30

 x = 30/2

 x = 15.

So, the value of ‘x’ is 15.


3. Solve 3x + 45 = 9x + 25.

Solution:

Transferring the variables and constants on the respective sides of the equation, we get,

   3x – 9x = 25 – 45

   -6x = -20

    x = 20/6

    x = 10/3.

So, the value of the variable, x = 10/3.


Forming linear equations in one variable from given word problem and solving them:

Following are the steps involved in formation of linear equation from the given word problem:

Step I: First of all read the given problem carefully and note down the given and required quantities separately.

Step II: Denote the unknown quantities as ‘x’, ‘y’, ‘z’, etc. 

Step III: Then translate the problem into mathematical language or statement.

Step IV: Form the linear equation in one variable using the given conditions in the problem.

Step V: Solve the equation for the unknown quantity.


Now let us try to form some linear equations from given word problems.

1. Sum of two numbers is 48. If one number is 5 times the other, find the numbers.

Solution:

Let one of the numbers be ‘x’. then second number is 5x.

Then, x + 5x = 48

               6x = 48

                x = 48/6

                x = 8.

So 1st number = 8.

2nd number = 5x = 5 x 8 = 40.


2. A total of $ 34,000is distributed as award prices among students. If the cash contains $ 100 and $ 500 noted in the ratio of 2 : 3. Then calculate the number of $ 100 and $ 500 notes that were distributed.

Solution:

Since we are being given about the ratio of $ 100 as well as $ 500 notes. 

So,

Let common ratio of number of notes be ‘x’. Then,

Number of $ 100 notes = 2x.

Number of $ 500 notes = 3x.

Total amount = 100 x 2x + 500 x 3x

                    = 200x + 1500x 

                    = 1700x

Since total amount distributed is $ 14,000.

So, 1700x = 14,000

             x = 14,000/1,700

             x = 20.

So, number of $ 100 notes = 2 × 20 = 40

     Number of $ 500 notes = 3 × 20 = 60.






9th Grade Math

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