Problems on Linear Equations in One Variable

Solved algebra problems on linear equations in one variable are explained below with the detailed explanation.

Let’s once again recall the methods of solving linear equations in one variable. 

 Read the linear problem carefully and note what is given in the question and what is required to find out. 

 Denote the unknown by any variable as x, y, ……. (any variable) 

 Translate the problem to the language of mathematics or mathematical statements. 

 Form the linear equation in one variable using the conditions given in the problems. 

 Solve the equation for the unknown. 

 Verify to be sure whether the answer satisfies the conditions of the problem. 

Worked-out problems on linear equations in one variable:

1. The sum of three consecutive multiples of 4 is 444. Find these multiples. 

Solution: 

If x is a multiple of 4, the next multiple is x + 4, next to this is x + 8. 

Their sum = 444

According to the question, 

x + (x + 4) + (x + 8) = 444 

⇒ x + x + 4 + x + 8 = 444

⇒ x + x + x + 4 + 8 = 444 

⇒ 3x + 12 = 444

⇒ 3x = 444 - 12 

⇒ x = 432/3 

⇒ x = 144

Therefore, x + 4 = 144 + 4 = 148 

Therefore, x + 8 - 144 + 8 – 152

Therefore, the three consecutive multiples of 4 are 144, 148, 152.



2. The denominator of a rational number is greater than its numerator by 3. If the numerator is increased by 7 and the denominator is decreased by 1, the new number becomes 3/2. Find the original number.

Solution:

Let the numerator of a rational number = x

Then the denominator of a rational number = x + 3

When numerator is increased by 7, then new numerator = x + 7

When denominator is decreased by 1, then new denominator = x + 3 - 1

The new number formed = 3/2

According to the question,

(x + 7)/(x + 3 - 1) = 3/2

⇒ (x + 7)/(x + 2) = 3/2

⇒ 2(x + 7) = 3(x + 2)

⇒ 2x + 14 = 3x + 6

⇒ 3x - 2x = 14 - 6

⇒ x = 8

The original number i.e., x/(x + 3) = 8/(8 + 3) = 8/11




3. The sum of the digits of a two digit number is 7. If the number formed by reversing the digits is less than the original number by 27, find the original number.

Solution:

Let the units digit of the original number be x.

Then the tens digit of the original number be 7 - x

Then the number formed = 10(7 - x) + x × 1

                                     = 70 - 10x + x = 70 - 9x

On reversing the digits, the number formed

                  = 10 × x + (7 - x) × 1

                  = 10x + 7 - x = 9x + 7

According to the question,

New number = original number - 27

⇒ 9x + 7 = 70 - 9x - 27

⇒ 9x + 7 = 43 - 9x 

⇒ 9x + 9x = 43 – 7

⇒ 18x = 36 

⇒ x = 36/18 

⇒ x = 2 

Therefore, 7 - x

              = 7 - 2

              = 5

The original number is 52


4. A motorboat goes downstream in river and covers a distance between two coastal towns in 5 hours. It covers this distance upstream in 6 hours. If the speed of the stream is 3 km/hr, find the speed of the boat in still water.

Solution:

Let the speed of the boat in still water = x km/hr.

Speed of the boat downstream = (x + 3) km/hr.

Time taken to cover the distance = 5 hrs

Therefore, distance covered in 5 hrs = (x + 3) × 5   (D = Speed × Time)

Speed of the boat upstream = (x - 3) km/hr

Time taken to cover the distance = 6 hrs.

Therefore, distance covered in 6 hrs = 6(x - 3)

Therefore, the distance between two coastal towns is fixed, i.e., same.

According to the question,

5(x + 3) = 6(x - 3)

⇒ 5x + 15 = 6x - 18

⇒ 5x - 6x = -18 – 15

⇒ -x = -33

⇒ x = 33

Required speed of the boat is 33 km/hr.



5. Divide 28 into two parts in such a way that 6/5 of one part is equal to 2/3 of the other.

Solution:

Let one part be x.

Then other part = 28 - x

It is given 6/5 of one part = 2/3 of the other.

⇒ 6/5x = 2/3(28 - x)

⇒ 3x/5 = 1/3(28 - x)

⇒ 9x = 5(28 - x)

⇒ 9x = 140 - 5x

⇒ 9x + 5x = 140

⇒ 14x = 140

⇒ x = 140/14

⇒ x = 10

Then the two parts are 10 and 28 - 10 = 18.




6. A total of $10000 is distributed among 150 persons as gift. A gift is either of $50 or $100. Find the number of gifts of each type.

Solution:

Total number of gifts = 150

Let the number of $50 is x

Then the number of gifts of $100 is (150 - x)

Amount spent on x gifts of $50 = $ 50x

Amount spent on (150 - x) gifts of $100 = $100(150 - x)

Total amount spent for prizes = $10000

According to the question,

50x + 100 (150 - x) = 10000

⇒ 50x + 15000 - 100x = 10000

⇒ -50x = 10000 - 15000

⇒ -50x = -5000

⇒ x = 5000/50

⇒ x = 100

⇒ 150 - x = 150 - 100 = 50

Therefore, gifts of $50 are 100 and gifts of $100 are 50.


The above step-by-step examples demonstrate the solved problems on linear equations in one variable.


 Equations

What is an Equation?

What is a Linear Equation?

How to Solve Linear Equations?

Solving Linear Equations

Problems on Linear Equations in One Variable

Word Problems on Linear Equations in One Variable

Practice Test on Linear Equations

Practice Test on Word Problems on Linear Equations


 Equations - Worksheets

Worksheet on Linear Equations

Worksheet on Word Problems on Linear Equation








7th Grade Math Problems 

8th Grade Math Practice 

From Problems on Linear Equations in One Variable to HOME PAGE




Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need.



New! Comments

Have your say about what you just read! Leave me a comment in the box below. Ask a Question or Answer a Question.




Share this page: What’s this?

Recent Articles

  1. Addition of Decimals | How to Add Decimals? | Adding Decimals|Addition

    Apr 17, 25 01:17 PM

    We will discuss here about the addition of decimals. Decimals are added in the same way as we add ordinary numbers. We arrange the digits in columns and then add as required. Let us consider some

    Read More

  2. Expanded form of Decimal Fractions |How to Write a Decimal in Expanded

    Apr 17, 25 12:21 PM

    Expanded form of Decimal
    Decimal numbers can be expressed in expanded form using the place-value chart. In expanded form of decimal fractions we will learn how to read and write the decimal numbers. Note: When a decimal is mi…

    Read More

  3. Math Place Value | Place Value | Place Value Chart | Ones and Tens

    Apr 16, 25 03:10 PM

    0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 are one-digit numbers. Numbers from 10 to 99 are two-digit numbers. Let us look at the digit 6 in the number 64. It is in the tens place of the number. 6 tens = 60 So…

    Read More

  4. Place Value and Face Value | Place and Face Value of Larger Number

    Apr 16, 25 02:55 PM

    Place Value of 3-Digit Numbers
    The place value of a digit in a number is the value it holds to be at the place in the number. We know about the place value and face value of a digit and we will learn about it in details. We know th…

    Read More

  5. Face Value and Place Value|Difference Between Place Value & Face Value

    Apr 16, 25 02:50 PM

    Place Value and Face Value
    What is the difference between face value and place value of digits? Before we proceed to face value and place value let us recall the expanded form of a number. The face value of a digit is the digit…

    Read More