When we solve a simple equation in algebra, we may have the following two situations.

It is possible that the equation may be true for all values of the variable. That is, the equation will have infinitely many solutions. Such an equation is called an identity.

It is also possible that the equation may have no solution. That is, there is no value of the variable that will result in a true equation.

When we solve a simple equation with infinitely many solutions or no solution, in the result we get at the last step, the variable will vanish. That is, there will be no variable.

If the result at the last step is true, then the equation has infinitely many solutions.

If the result at the last step is false, then the equation has no solution.

We can understand this stuff more clearly in the following examples.

**Example 1 : **

Solve the following equation :

2(1 - x) + 5x = 3(x + 1)

**Solution : **

2(1 - x) + 5x = 3(x + 1)

Simplify both sides.

2 - 2x + 5x = 3x + 3

2 + 3x = 3x + 3

Subtract 3x from each side.

2 = 3

The above result is false. Because 2 is not equal to 3. Because the result we get at the last step is false, the given equation has no solution.

**Example 2 : **

Solve the following equation :

(1/2)(8y - 6) = 5y - (y + 3)

**Solution : **

(1/2)(8y - 6) = 5y - (y + 3)

Simplify both sides.

4y - 3 = 5y - y - 3

4y - 3 = 4y - 3

Subtract 4y from each side.

-3 = -3

The above result is true. Because the result we get at the last step is true, the given equation has infinitely has many solutions.

**Example 3 : **

Solve the following equation :

(1/3)(9 - 6x) = 5 - 2x

**Solution : **

(1/3)(9 - 6x) = 5 - 2x

Simplify.

3 - 2x = 5 - 2x

Add 2x to each side.

3 = 5

The above result is true. Because the result we get at the last step is true, the given equation has infinitely has many solutions.

**Example 4 : **

(1/3)(15 - 6x) = 5 - ax

If the linear equation above is an identity, what is the value of a ?

**Solution : **

(1/3)(15 - 6x) = 5 - ax

Simplify.

5 - 2x = 5 - ax

Because the given equation is an identity, the coefficients of like terms on both sides must be equal.

That is, coefficients of 'x' terms on the left side and right side must be equal.

So, equate the coefficients of 'x'.

- 2 = - a

Multiply each side by (-1).

2 = a

Therefore, the value of a is 2.

Apart from the stuff given in this section, 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 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**