**Word Problems Involving Linear Equations :**

In this section, we will learn, how to solve word problems using linear equations.

There is a simple trick behind solving word problems using linear equations.

The picture shown below tells us the trick.

**Example 1 :**

Of two numbers, 1/5th of a the greater equal to 1/3rd of the smaller and their sum is 16. Find the numbers.

**Solution :**

Let "x" and "y" be the two numbers such that

x > y

**Given :** The sum of the two number is 16.

So, we have

x + y = 16 ------(1)

**Given :** 1/5th of a the greater equal to 1/3rd of the smaller

So, we have

1/5 ⋅ x = 1/3 ⋅ y

Simplify.

3x = 5y ------(2)

Solving (1) and (2), we get

x = 10 and y = 6

Hence, the two numbers are 10 and 6.

**Example 2 :**

The wages of 8 men and 6 boys amount to $33. If 4 men earn $4.50 more than 5 boys, determine the wages of each man and boy.

**Solution :**

Let "x" and "y" be the wages of each man and boy.

**Given :** The wages of 8 men and 6 boys amount to $33.

So, we have

8x + 6y = 33 ------(1)

**Given :** 4 men earn $4.50 more than 5 boys.

So, we have

4x - 5y = 4.50 ------(2)

Solving (1) and (2), we have

x = 3 and y = 1.5

Hence, the wages of each man and each boy are $3 and $1.50 respectively.

**Example 3 :**

A number between 10 and 100 is five times the sum of its digits. If 9 be added to it the digits are reversed. Find the number.

**Solution :**

Because the number is between 10 and 100, it has to be a two digit number.

So, let "xy" be the number between 10 and 100.

**Given : ** The number between 10 and 100 is five times the sum of its digits.

xy = 5(x + y) ------(1)

xy is a two digit number.

And x is in tens place and y is in ones place.

So, we have

xy = 10 ⋅ x + 1 ⋅ y

xy = 10x + y

Then, we have

(1)------> 10x + y = 5(x + y)

10x + y = 5x + 5y

Simplify.

5x - 4y = 0 -------(2)

**Given : ** If 9 be added to it the digits are reversed.

xy + 9 = yx

10 ⋅ x + 1 ⋅ y + 9 = 10 ⋅ y + 1 ⋅ x

10x + y + 9 = 10y + x

Simplify.

9x - 9y = - 9

Divide both sides by 9.

x - y = - 1 ------(3)

Solving (2) and (3), we get

x = 4 and y = 5

Hence, the required number is 45.

**Example 4 :**

The age of a man is three times the sum of the ages of his two sons and 5 years hence his age will be double the sum of their ages. Find the present age of the man.

**Solution :**

Let "x" be the present age of the man and "y" be the sum of the present ages of his two sons. **Given :** Present age of the man is 3 times the sum of the ages of 2 sons.

So, we have

x = 3y ------(1)

5 years hence,

Age of the man = x + 5

Sum of the ages of his two sons = y + 5 + 5 = y + 10

(There are two sons in y. So 5 is added twice)

**Given :** 5 years hence, age of the man will be double the sum of the ages of his two sons.

So, we have

x + 5 = 2(y + 10)

Simplify.

x + 5 = 2y + 20

x = 2y + 15 ------(2)

From (1), we can plug x = 3y in (2).

(2)------> 3y = 2y + 15

Subtract 2y from both sides.

y = 15

Plug y = 15 in (1).

(1)------> x = 3 ⋅ 15

x = 45

Hence the present age of the man is 45 years.

**Example 5 :**

A trader has 100 units of a product. He sells some of the units at $6 per unit and the remaining units at $8 per units. He receives a total of $660 for all 100 units. Find the number units sold in each category.

**Solution :**

Let x be the no. of units sold at $6/unit and y be the no. of units sold at $8/unit.

**Given :** The trader sells 100 units in all.

So, we have

x + y = 100 ------(1)

**Given :** He receives a total of $660 for all 100 units.

So, we have

6x + 8y = 660

Divide both sides by 2.

3x + 4y = 330 ------(2)

Solving (1) and (2), we get

x = 70 and y = 30

Hence, the no. of tickets sold at $6 per unit is 70 and the no. of tickets sold at $8 per unit is 30.

After having gone through the stuff given above, we hope that the students would have understood, how to solve word problems involving involving linear equations.

Apart from the stuff, if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

You can also visit our following web pages on different stuff in math.

**WORD PROBLEMS**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Trigonometry word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**

**Sum of all three four digit numbers formed using 0, 1, 2, 3**

**Sum of all three four digit numbers formed using 1, 2, 5, 6**

HTML Comment Box is loading comments...