Equations

In every mathematical equation, the formula adheres to a fundamental structure: LHS (Left Hand Side) equals RHS (Right Hand Side). This balance is what defines an equation. The purpose of solving equations is to find the values of unknown variables that make the equation true. Without an equals sign, the statement does not qualify as an equation but is instead considered an expression. The distinctions between equations and expressions will be elaborated upon further in the subsequent sections of this article.

What are Equations?

Equations are mathematical statements that assert the equality of two expressions. They are composed of two expressions on either side of an equals sign. The primary purpose of an equation is to represent a relationship where two values are equal under certain conditions.

Parts of an Equation

img

An equation is a fundamental concept in mathematics, used to express the equality between two expressions. Understanding the different parts of an equation is crucial for effectively solving them. Here’s a breakdown of the essential components:

1. Expressions

Left-Hand Side (LHS): This is the expression to the left of the equals sign. It represents one half of the equation.Right-Hand Side (RHS): Opposite the LHS, this expression is located to the right of the equals sign.

2. Equals Sign (=)

The equals sign is a critical component that differentiates an equation from an expression. It indicates that the values on either side are equivalent in terms of mathematical value.

3. Variables

Variables are symbols that represent unknown quantities within an equation. They are placeholders that can be solved by manipulating the equation.

4. Constants

Constants are fixed values that do not change. In equations, these are typically represented by numbers and are used to help define the relationship between the variables.

5. Coefficients

Coefficients are numbers used to multiply a variable. They quantify the contribution of the variable to the equation.

6. Operators

Operators are symbols that perform operations on the variables and constants. Common operators include addition (+), subtraction (−), multiplication (×), and division (÷).

7. Terms

A term can be a variable, a constant, or a combination of both multiplied together. An equation is generally composed of multiple terms.

Types of Equations

Equations are foundational in mathematics, allowing us to express relationships between variables and constants. Understanding different types of equations can help in solving specific mathematical problems. Here’s an overview of the major types:

1. Linear Equations

• Characteristics: These equations form a straight line when graphed. They have no variables raised to a power higher than one.
• Example: 𝑦=3𝑥+4y=3x+4

2. Quadratic Equations

• Characteristics: Quadratic equations involve a variable squared (𝑥2x2) and form a parabolic curve when graphed.
• Example: 𝑎𝑥2+𝑏𝑥+𝑐=0ax2+bx+c=0

3. Cubic Equations

• Characteristics: Involving a variable raised to the third power (𝑥3x3), these equations can have one to three real roots.
• Example: 𝑥3−9𝑥=0x3−9x=0

4. Polynomial Equations

• Characteristics: These involve variables raised to positive integer powers. The highest power of the variable is called the degree of the polynomial.
• Example: 4𝑥4−3𝑥2+7𝑥−2=04x4−3x2+7x−2=0

5. Rational Equations

• Characteristics: These equations feature at least one fractional expression involving a polynomial in the numerator and denominator.
• Example: 𝑥−1𝑥+2=3x+2x−1​=3

6. Differential Equations

• Characteristics: Involve functions and their derivatives. These are fundamental in modeling dynamic systems.
• Example: 𝑑𝑦𝑑𝑥=3𝑦dxdy​=3y

7. Exponential Equations

• Characteristics: These equations have variables in the exponent.
• Example: 2𝑥=82x=8

8. Logarithmic Equations

• Characteristics: Equations that involve logarithms.
• Example: log⁡(𝑥)+log⁡(𝑥−3)=1log(x)+log(x−3)=1

9. Trigonometric Equations

• Characteristics: These involve trigonometric functions such as sine, cosine, and tangent.
• Example: sin⁡(𝑥)=22sin(x)=22​​

10. Absolute Value Equations

• Characteristics: Equations that involve the absolute value function, which measures the distance of a number from zero on the number line.
• Example: ∣2𝑥−5∣=3∣2x−5∣=3

11. System of Equations

• Characteristics: Consists of multiple equations to be solved together, where the solution is the set of values that satisfies all equations simultaneously.
• Example:
  • 𝑥+𝑦=10x+y=10
  • 2𝑥−𝑦=12x−y=1

How to Solve an Equation?

Solving equations is a fundamental skill in mathematics, essential for finding unknown values that satisfy the equation. Here’s a step-by-step guide on how to approach solving equations:

Step 1: Simplify Both Sides

• Combine like terms on each side of the equation. Like terms are terms that have the same variables raised to the same power.
• Eliminate parentheses using the distributive property if necessary.

Step 2: Isolate the Variable

• Decide which variable to solve for if there are multiple variables.
• Use basic arithmetic operations to isolate the variable on one side of the equation. This might involve adding, subtracting, multiplying, or dividing both sides of the equation by the same number or expression.

Step 3: Perform Operations to Isolate the Variable

• If the variable is multiplied by a coefficient, divide both sides by that coefficient to solve for the variable.
• If the variable has a term added or subtracted, do the opposite operation on both sides to move that term to the opposite side of the equation.

Step 4: Check Your Solution

• Substitute the solution back into the original equation to ensure that both sides are equal. This verifies whether the solution is correct.
• Simplify the equation after substitution to confirm equality.

Step 5: State the Solution

• Clearly state the value of the variable that solves the equation. If the equation is true for all values of the variable, state that the equation is an identity. If no value satisfies the equation, indicate that it has no solution.

Equation vs Expression

In algebra, expressions and equations are fundamental concepts that are often used simultaneously but represent different mathematical ideas. Here’s a clearer distinction between the two through a descriptive table:

img

Important Notes on Equations in Math: