**Reflexive Property :**

For any real number a,

a = a

**Symmetric Property :**

For any two real numbers a and b, if a = b, then b = a.

That is,

a = b -----> b = a

**Transitive Property :**

For any three real numbers a, b and c, if a = b and b = c, then a = c.

That is,

a = b, b = c -----> a = c

**Substitution Property :**

If a = b, then a can be substituted for b or b can be substituted for a in any equation or expression.

**Addition Property :**

Two sides of an equation remain equal, if the same number is added to each side.

For example, if a = b, then

a + c = b + c

**Subtraction Property :**

**Two sides of an equation remain equal, if the same number is subtracted from each side.**

For example, if a = b, then

a - c = b - c

**Multiplication Property :**

**Two sides of an equation remain equal, if each side is multiplied by the same number. **

For example, if a = b, then

a**⋅**c = b**⋅**c

**Division Property :**

**Two sides of an equation remain equal, if each side is divided by the same number. **

For example, if a = b, then

a ÷ c = b ÷ c

**Distributive Property of Multiplication over Addition :**

**For all real numbers a, b and c, **

a(b + c) = a ⋅ b + a ⋅ c

**Distributive Property of Multiplication over Subtraction :**

**For all real numbers a, b and c, **

a(b - c) = a ⋅ b - a ⋅ c

**Commutative Property of Addition :**

**For all real numbers a an b, **

a + b = b + a

**Commutative Property of Multiplication :**

**For all real numbers a an b, **

a ⋅ b = b ⋅ a

**Associative Property of Addition :**

**For all real numbers a, b and c, **

a + (b + c) = (a + b) + c

**Associative Property of Multiplication :**

**For all real numbers a, b and c, **

a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c

**Note : **

Commutative and Associative Properties are not true for subtraction and division.

Property of equality along with other properties from algebra, such as the distributive property,

a(b + c) = ab + ac

can be used to solve equations.

For instance, let us solve the equation given below.

3(x + 2) = 2x + 8

Apply Distributive property on the left side of the equation.

3x + 6 = 2x + 8

Subtraction property :

Subtract 2x from each side of the equation.

x + 6 = 8

Subtract 6 from each side of the equation.

x = 2

**Problem 1 : **

Solve 7x - 2 = 4x + 9 and write a reason for each step.

**Solution : **

Given :

7x - 2 = 4x + 13

Subtraction Property of Equality :

Subtract 4x from each side.

3x - 2 = 13

Addition Property of Equality :

Add 2 to each side.

3x = 15

Division Property of Equality :

Divide both sides by 2.

x = 5

**Problem 2 : **

Solve 52y - 3(12 + 9y) = 64 and write a reason for each step.

**Solution : **

Given :

52y - 3(12 + 9y) = 64

Distributive Property :

Distribute 3 to 12 and 9y.

52y - 36 - 27y = 64

Simplify :

25y - 36 = 64

Addition Property of Equality :

Add 36 to each side.

25y = 100

Division Property of Equality :

Divide both sides by 25.

y = 4

**Problem 3 : **

When we do exercise every day, we should find our target heart rate. This is the rate at which we achieve an effective workout while not placing too much strain on our heart. Our target heart rate r (in beats per minute) can be determined from our age a (in years) using the equation a = 220 - 10r/7.

(i) Solve the formula for r and write a reason for each step.

(ii) Use the result to find the target heart rate for a 16 year old.

(iii) Find the target heart rate for the following ages :

20, 30, 40, 50 and 60

What happens to the target heart rate as a person gets older ?

**Solution (i) : **

Given :

a = 220 - 10r/7

Addition Property of Equality :

Add 10r/7 to each side.

a + 10r/7 = 220

Subtraction Property of Equality :

Subtract a from each side.

10r/7 = 220 - a

Multiplication Property of Equality :

Multiply both sides by 7/10.

r = 7/10 ⋅ (220 - a)

**Solution (ii) : **

To find the target heart rate for a 16 year old, substitute a = 16.

r = 7/10 ⋅ (220 - 16)

Simplify :

r = 7/10 ⋅ 204

r = 1428 / 10

r = 142.8

The target hear rate for a 16 year old is about 142.8 beats per minute.

**Solution (iii) : **

The table given below shows the heart rate for ages 20, 30, 40, 50 and 60.

From the above table, it is clear that the target heart rate appears to decrease as a person gets older

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

If you have any feedback about our math content, please mail us :

**v4formath@gmail.com**

We always appreciate your feedback.

You can also visit the 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**