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When two mathematical expressions are equated with an "=" sign, we get a math equation. One or both sides of the equation may contain variables. By solving a math equation, we are trying to find a single value which satisfies the math equation. That is when we substitute the value of the solution for the variable, the equation balances.

When two mathematical expressions are equated with an "=" sign, we get a math equation. One or both sides of the equation may contain variables. By solving a math equation, we are trying to find a single value which satisfies the math equation. That is when we substitute the value of the solution for the variable, the equation balances.

## Steps to solve math equation

- Linear equation
- Equation containing radicals
- Equations containing absolute values
- Quadratic equations
- Equations containing fractions
- Exponential equation
- Logarithmic equation

## Solve any Math Equation

**Example 1: -**

Solve the equation 5x - 4 = 2x - 6

**Solution: -**

Given equation is 5x - 4 = 2x - 6

Here there is only one variable involved in the problem. So to solve, we just have to isolate the variable. For this we take all variables to one side and all constants to other.

Lets first take the constants to the right side of the equation. For this, we add 4 on both sides, we get

5x = 2x -2

Now we take the variables on the left side of the equation. For this, we subtract 2x on both sides, we get

3x = 2

Now we have variable on the left side and constant on the right side of the equation

Dividing by 3 on both sides, we get the required solution

x = 2/3

**Solution: x = 2/3**

**Example 2: -**

Solve the equation (x - 1)(x - 3 )(x - 5)(x - 7) = 9

**Solution: -**

Given equation is (x - 1)(x - 3 )(x - 5)(x - 7) = 9

Here there is only one variable involved in the problem. So to solve, we just have to isolate the variable. First we rearrange the factors.

So the equation becomes

(x - 1)(x - 7 )(x - 3)(x - 5) = 9

Now we combine the terms, we get

(x

^{2}- 8x + 7)(x

^{2}- 8x +15) = 9

Step 1:

Step 1:

Take x

^{2}- 8x as u. Then the equation becomes (u + 7)(u + 15) = 9

Simplifying the left side we get

u

^{2}+ 7u + 15u + 105 = 9

That is

u

^{2}+ 22u + 105 = 9

subtracting 9 on both sides, we get

u

^{2}+ 22u + 96 = 0

Now we split the middle term

u

^{2}+ 16u + 6u + 96 = 0

Taking u common from first two terms and 6 common from last two terms, we get

u (u + 16) + 6 (u + 16) = 0

Now we take u + 16 common, we get

(u + 16) ( u + 6) = 0

Equating each factor to 0, we get

u + 16 = 0 ⇒ u = -16

and u + 6 = 0 ⇒ u = -6

**Step 2:**

When u = -16

x

^{2}- 8x = -16

Adding 16 on both sides, we get

x

^{2}- 8x + 16 = 0

Now we split the middle term

x

^{2}- 4x - 4x + 16 = 0

Taking x common from first two terms and -4 common from last two terms, we get

x (x - 4) - 4 (x - 4) = 0

Now we take x - 4 common, we get

(x - 4) ( x - 4) = 0

Equating each factor to 0, we get

x - 4 = 0 ⇒ x = 4

and x - 4 = 0 ⇒ x = 4

**Step 3:**

When u = -6

x

^{2}- 8x = -6

Adding 6 on both sides, we get

x

^{2}- 8x + 6 = 0

We use the quadratic formula to solve this equation.

x = [-(-8) ±√((-8)

^{2}- 4 x 1 x 6)] / 2

= (8 ±√40)/2

= 4 ±√10

**Therefore solutions are 4, 4, 4±√10**

**Example 3: -**

Solve the equation log(x - 9) + log x = 1

**Solution: -**

Given equation is log(x - 9) + log x = 1

Here there is only one variable involved in the problem. So to solve, we just have to isolate the variable.First we have to get rid of the log

We know that log a + log b = log ab

So the left side becomes

log (x - 9)x = 1

Also we know that a

^{x}= b ⇒ log

_{a}b = x

So we can write the above equation as

(x - 9)x = 10

^{1}That is x

^{2}- 9x = 10.

This is quadratic equation. Lets solve it by grouping.

Subtracting 10 on both sides, we get

x

^{2}- 9x - 10 = 0

Now we split the middle term

x

^{2}- 10x + x -10 = 0

Taking x common from first two terms and 1 common from last two terms, we get

x (x - 10) + 1 (x - 10) = 0

Now we take x - 10 common, we get

(x - 10) ( x + 1) = 0

Equating each factor to 0, we get

x - 10 = 0 ⇒ x = 10

and x + 1 = 0 ⇒ x = -1

When we substitute these values of x in the equation, we can see that -1 is not a solution

**Therefore solution is : x = 10**