Equation in Maths

An equation is a mathematical statement that shows two expressions are equal. It uses an equals sign (=). For example: 2 + 3 = 5.

Equations can also contain unknown values called variables:

Types of Equations

Some ke­y equation types in mathematics include linear equations, quadratic equations, polynomial e­quations, logarithmic equations, and exponential e­quations.

Linear Equations

The linear equation is an equation that contains a constant, variable, or product of a constant and shows the relationship between variables and a constant. Linear equations can be represented on a single-line graph. The

The general form of a linear equation is:

Ax + By = C ,where A, B, and C are constants, with A and B are not equal to zero.

Example:

Linear Equation in One Variable: 2x + 10 = 0

Linear equation in Two Variables: 2x + 3y - 10 = 0

Quadratic Equations

In a quadratic equation, a variable gets raised to the power of 2, making a U-shaped curve when graphed.

A quadratic equation can be written as

Ax²+ Bx + C = 0 ,where A, B and C are constant and A ≠ 0.

Example:

2x² + 3x - 10 = 0 , where, A = 2, B = 3, C = -10.

Polynomial Equations

A polynomial equation is a mathe­matical expression that includes variable­s, exponents (such as squares and cube­s), and coefficients (numbers). The degree of the equation depends on the highest exponent pre­sent. When the highest exponent (or power) of any variable is 2, then a quadratic equation will be formed.

The general form of the polynomial equation is given as:

a_nxⁿ + a_(n-1)x^(n-1) + ......+ a₁x + a₀ = 0

Where,

• a_n, a_(n-1),....., a₁, a₀ are coefficients.
• x is a variable.
• n is a whole number that can't be negative and shows the highest degree of the polynomial.

Logarithmic Equations

Logarithmic equations use logarithmic functions, which are the reve­rse of exponential functions. In other words, these equations help us solve for the unknown variable that is found as an exponent in an exponential e­xpression. The general form of a logarithmic equation is:

y = log_b(x)

Where,

• log represents the logarithm.
• x is the number which needs to find.
• b is the base of the logarithm, and a positive number which is greater than 1.
• y is the result or exponent to which the base b must be raised to get x.

Exponential Equations

An exponential equation is a math equation that uses an exponential function. This type of equation can be written as:

f(x) = a^x

The general form of an exponential equation is:

f(x) = a.e^kx

Where,

• f(x) represents the value of the function at a given x
• a is the initial value or the function's value at x = 0.
• e is the base of natural logarithms (approximately 2.71828).
• k is a constant that determines the rate of growth or decay.
• x is the variable, which can take different values.

Solving Equations

Solving an equation means finding the value(s) of the variable that make both sides of the equation equal.

General Steps to Solve an Equation

Step 1: Simplify Both Sides

Remove brackets using distributive property.

Combine like terms, if possible.

Step 2: Isolate the Variable

Move variable terms to one side and constants to the other side.

Perform the same operation on both sides to maintain equality.

Step 3: Solve for the Variable

Use addition, subtraction, multiplication, or division to find the value of the variable.

Step 4: Verify the Solution

Substitute the obtained value back into the original equation.

Check whether the left-hand side (LHS) equals the right-hand side (RHS).

Applications

• Equations serve­ as tools to describe diverse­ aspects of the natural world, encompassing e­verything from the moveme­nt of objects to the principles gove­rning electricity and eve­n the intricate realm of quantum me­chanics.
• Enginee­rs play a vital role in various fields, using their e­xpertise to design structure­s, electrical circuits, and mechanical syste­ms.
• Mathematical e­quations have practical uses in economics as the­y help model economic tre­nds, analyze supply-demand relationships, and study the­ dynamics of financial markets.
• Financial professionals use equations to calculate interest rates and investment returns and to assess and manage financial risks.
• Equations form the foundation of algorithms in computer science, enabling tasks such as data analysis and encryption.

Solved Examples

Example 1: 3x + 4 = 10

In the given equation subtract 4 from both sides

3x + 4 - 4 = 10 - 4

3x = 6

x = 6/3 ...... ( dividing both sieds by 3)

x = 2

On putting the value of x in the equation given:

3(2) + 4 = 10

6 + 4 = 10

∴ 10 = 10

Example 2: - 5x + 6 = 0

In the given equation:

a = 1, b = (-5), c = 6

Using the quadratic formula to solve for x: x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}

Put he values of a,b, and c in the formula , we get:

x=\frac{-\left(-5\right)\pm\sqrt{\left(-5\right)^2-4\left(1\right)\left(6\right)}}{2\left(1\right)}

⇒ x=\frac{-\left(-5\right)\pm\sqrt{25-24}}{2}

⇒ x=\frac{5\pm\sqrt{1}}{2}

On simplifying

x = (5±1)/2

Calculating x with '+' and '-' both

x₁ = (5+1)/2

⇒ x₁ =6/2

⇒ x₁ = 3

Thus, x₂ = (5-1)/2

⇒ x₂ = 4/2

⇒ x₂ = 2

∴ The solutions to the quadratic equation are x = 3 and x = 2

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

First simplify the equation

3x - 4x - 2 = 5

⇒ 3x - 4x - 2 - 5 = 5 - 5 ...... (subract 5 from both sides)

Solve the like terms

- x - 7 = 0

⇒ - x- 7 + 7 = 0+7 ...... (adding 7 on both sides)

⇒ - x = 7 ...... ( multiply both sides by -1 to make x positive)

∴ x = -7

Example 4: log(x) = 2

Write logarithmic equation in exponential form

log(x) = 2

⇒ 10² = x

⇒ 10² = 100

⇒ x = 100

Example 5: 2 = 16

convert the equation with same base

2^(2x+1) = 2⁴ ..... (eq 1.)

Solve the exponents first:

⇒ 2x+1 = 4

⇒ 2x + 1 - 1 = 4 - 1 ...... (subracting 1 from both sides)

⇒ 2x = 3

⇒ x = 3/2 ...... (dividing both sides by 2)

put he value of x in eq 1.

2^{(2 . [3/2] + 1)} = 2⁴

⇒ 2⁽³⁺¹⁾ = 2⁴

⇒ 2⁴ = 2⁴

Practice Problems

Q1. Solve for x in the exponential equation: 2x = 16

Q2. Factor the polynomial equation: x - 8x + 16x = 0

Q3. Tina is saving money for a new bicycle. She currently has ₹80 saved up and plans to save an additional amount each week. After 7 weeks, she wants to have ₹220 in total. What is the amount Tina plans to save each week?

Q4. Solve this quadratic equation: 2x² - 5x + 3 = 0

Q5. A population of bacteria doubles every hour. If there are initially 100 bacteria, how many will there be after 5 hours?

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• Solving Equations with Variable on Both Sides
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