91Ó°ÊÓ

When we solve equations in mathematics, we are essentially finding the values of the variables that make the equation true. With any equation, the key is to isolate the variable on one side to determine its value. Here’s how we can do this step-by-step:

1. **Identify and Simplify:** First, make sure both sides of the equation are as simple as possible. Combine like terms and remove any parentheses using distribution.
2. **Move Terms:** Get all variable terms on one side of the equation and constant terms on the other. This is often done through addition or subtraction.
3. **Isolate the Variable:** Once the variable is on one side, use multiplication or division to solve for the variable, as we did by isolating \( p \) by adding 1 and then dividing by 3.
Breaking down an equation into smaller, manageable steps can effectively find the solution.

Logarithms are the inverse operations of exponentials, similar to how subtraction is the inverse of addition. A logarithm answers the question: to what power must a base be raised, to yield a certain number? This can be expressed as:

1. **Base of the Logarithm:** The base is the number that is raised to a power. In our exercise, the base is 7.
2. **Logarithm Rule:** When two logarithms with the same base are set equal, it implies the quantities inside them are also equal: if \( \log_b{x} = \log_b{y} \), then \( x = y \).
3. **Simplifying:** Understanding how to "remove" the logarithm using the same base makes it easier to solve equations as seen when \( \log_{7}(3p - 1) = \log_{7}9 \) is simplified to \( 3p - 1 = 9 \).
Logarithms are a powerful tool for solving equations involving exponential growth or decay.

Algebraic equations are statements where two expressions are set equal to each other. They can involve constants, coefficients, variables, and mathematical operations such as addition, subtraction, multiplication, and division. Here are a few core ideas:

1. **Variables and Constants:** A variable like \( p \) in our equation represents an unknown value, while constants are known values like 9 in our example.
2. **Coefficients:** These are numbers multiplied by variables. In \( 3p - 1 = 9 \), the number 3 is a coefficient.
3. **Balancing Equations:** Both sides of an equation must remain equal. This is why any operation performed on one side must also be performed on the other. For example, adding 1 to both sides.
Understanding these concepts allows us to manipulate equations to find unknowns, a fundamental skill in algebra.