SOLVING EQUATIONS

Solving an equation means to find all the values of the variable which make the equation true. The way to find the value of the variable is to isolate the variable that has a coefficient of 1 onto one side of the equation.

For example, consider the following equation in the variable 'x'.

2x + 3 = 0

Here, the variable x is multiplied by 2 and 3 is added to 2x. To isolate the variable x, we have to get rid of all the values around the variable x. We can do this using the rules of algebra called properties of equality.

Example 1 :

Solve :

x + (-11) = -25

Solution :

x + (-11) = -25

Add 1 to both sides.

x + (-11) + 11 = -25 + 11

x = -14

Example 2 :

Solve :

8y = -24

Solution :

8y = -24

Divide both sides by 8.

8y/8 = -24/8

y = -3

Example 3 :

-11 + y = 9

Given the above equation, find the value of 

20 - (11 - y)

Solution :

-11 + y = 9

Add 11 to both sides.

y = 20

The value of 20 - (11 - y) :

20 - (11 - y) = 20 - (11 - 20)

= 20 - (-9)

= 20 + 9

= 29

Example 4 :

If 33 - x = x + 27 - 5x, what is the value of (33 + 3x)?

Solution :

33 - x = x + 27 - 5x

33 - x = 27 - 4x

Add 5x to both sides.

33 + 3x = 27

Subtract 33 from both sides.

3x = -6

Divide both sides by 3.

x = -2

The value of (33 + 3x) :

33 + 3x = 33 + 3(-2)

= 33 - 6

= 27

Example 5 :

If (1/2)x + 3 = 3/4 - x, what is the value of x?

Solution :

(1/2)x + 3 = 3/4 - x

Add x to both sides.

x/2 + x + 3 = 3/4

Subtract 3 from both sides.

x/2 + x = 3/4 - 3

(x + 2x)/2 = (3 - 12)/4

3x/2 = -9/4

Multiply both sides by 2.

3x = -18/4

3x = -9/2

Divide both sides by 3.

x = -9/6

x = -3/2

Example 6 :

Solve the following equation :

x - (3 - 2x) + (4 - 5x) = -7

Solution :

x - (3 - 2x) + (4 - 5x) = -7

x - 3 + 2x + 4 - 5x = -7

-2x + 1 = -7

Subtract 1 from both sides.

-2x = -8

Divide both sides by -2.

x = 4

Example 7 :

(4/5)(x - 5) - (1/5)(x - 10) = 19

Solution :

(4/5)(x - 5) - (1/5)(x - 10) = 21

Multiply both sides of the equation by 5 to get rid of the denominators.

4(x - 5) - 1(x - 10) = 95

Use Distributive property.

4x - 20 - x + 10 = 95

3x - 10 = 95

Add 10 to both sides.

3x = 105

Divide both sides by 3.

x = 35

Example 8 :

If three quarters of a number decreased by twenty is equal to eighty two, what is that number?

Solution :

Let x be the number.

(3/4)x - 20 = 82

3x/4 - 20 = 82

Add 20 to both sides.

3x/4 = 102

Multiply both sides by 4.

3x = 408

Divide both sides by 3.

x = 136

The number is 136.

Example 9 :

There are one hundred forty-two students in a high school band. These students represent two ninth of the total students in the high school. Find the total number of students in the school.

Solution :

Let x be the total number of students in the school.

(2/9)x = 142

Multiply both sides by 9.

2x = 1278

Divide both sides by 2.

x = 639

The total number of students in the school is 639.

Example 10 :

The quotient of a number and five equals nine less than one half of the number. What is the number?

Solution :

Let x be the number.

x/5 = (1/2)x - 9

x/5 = x/2 - 9

In the equation above, we find two different denominators 5 and 2. 

The least common multiple of (2, 5) is 10.

Multiply both sides of the above equation by 10 to get rid of the denominators.

10(x/5) = 10(x/2 - 9)

2x = 10x/2 - 90

2x = 5x - 90

Subtract 5x from both sides.

-3x = -90

Divide both sides.

x = 30

The number is 30.

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