Lesson HOW TO algebreze and to solve word problems on numbers?

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How to algebreze and to solve word problems on numbers?


This lesson teaches you,  by examples,  how to write the expressions and the equations for word problems on numbers.

Problem 1

Laura is thinking of a number such that the sum of the number and five times two more than the number is 26 more than four times the number.
Determine the number Laura is thinking of.

Solution

Let  x  be the unknown number.

Then "the sum of the number and five times two more than the number"  is  x + 5*(x+2).
"26 more than four times the number"  is  4x + 26.

Thus the equation is

x + 5*(x+2) = 4x + 26.

Solve it:

x + 5x + 10 = 4x + 26.

6x - 4x = 26 - 10,

2x = 16,

x = 16%2F2 = 8.

Answer.  Laura is thinking of a number  8.


Problem 2

Twice the differences between a number,  x,  and fifteen is three times the number.  Find the number.

Solution

"Twice the differences between a number,  x,  and fifteen"  is  2*(x-15).
"Three times the number"  is  3x.

Therefore,  you have an equation

2*(x-15) = 3x.

2x - 30 = 3x,

3x - 2x = -30,

x = -30.

Answer. The number is -30.


Problem 3

Write an algebraic expression for this sentence:  ten more than twice the absolute value of a number.

Solution

2*|x| + 10.


Problem 4

Write an algebraic equation for this sentence:  14 less than 3 times a number is 5 more than half of the number.

Solution

"14 less than 3 times a number"  is  3x - 14.
"5 more than half of the number"  is  x%2F2 + 5.

Therefore,  an equation is

3x - 14 = x%2F2 + 5.


Problem 5

Write an algebraic expression for this sentence:  the sum of seven more than twice a number and the number.

Solution

(2x + 7) + x


Problem 6

Write an algebraic equation for this sentence:  three times the sum of a number and 3 is the same as five times the number divided by 7.

Solution

3*(x+3) = 5x%2F7.


Problem 7

Write an algebraic equation for this sentence:  the sum of a number and 3,  multiplied by the sum of the number and 8 is 19.

Solution

(x + 3)*(x + 8) = 19.


Problem 8

Write an algebraic equation for this sentence:  three times a number added to 4 times a number is the same as six times the number minus 6

Solution

3x + 4x = 6x - 6.


Problem 9

Write an algebraic equation for this sentence and solve it:  the difference of five times a number and 5 results in twice the difference of three times the number and 3.

Solution

According to the condition,  an equation for the unknown number  x  is

5x - 5 = 2*(3x - 3).

Solve it:

5x - 5 = 6x - 6

5x - 6x = -6 + 5,

-x = -1,

x = 1.

Answer. x = 1.


Problem 10

Five times the sum of twice a number and 5 is the same as the sum of 9 times the number and 5.  Find the number.

Solution

The equation is

5*(2x + 5) = 9x + 5.

Solve it:

10x + 25 = 9x + 5,

10x - 9x = 5-25 = -20.

x = -20.

Check:  The left side is  5*(2x + 5) = 5*(2*(-20) + 5) = 5*(-40 + 5) = 5*(-35) = -175.

             The right side is  9x + 5 = 9*(-20) + 5 = -180 + 5 = -175.

Answer. x = -20.


Problem 11

The sum of two numbers is 90.  The larger number is 14 more than 3 times the smaller number.  Find the numbers.

Solution

Let  x  be the smaller number.
The the other number is  3x + 14.  We know that their sum is  90.  It gives an equation

x + (3x + 14) = 90.

Solve it step by step:

x + 3x + 14 = 90,

4x = 90 - 14,

4x = 76,

x = 76%2F4 = 19.   It is the smaller number.

The other number is   90 - 19 = 71.

(it is really in  14  more than  3x:   3*19 + 14 = 57 +14 = 71).


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Use this file/link  ALGEBRA-I - YOUR ONLINE TEXTBOOK  to navigate over all topics and lessons of the online textbook  ALGEBRA-I.

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